Lower bounds for diophantine approximations

JOURNAL OF PURE AND APPLIED ALGEBRA

Journal of Pure and Applied Algebra 117 & 118 (1997) 277-3 17

ELSEVIER

Lower bounds for diophantine M. Giusti a,*, J. Heintzb, L.M.

approximations’

K. Htigeleb,

Pardob,

J.L.

J.E.

Moraisb,

Montaiia”

il GAGE, Centre de MathCmatiques, .&ColePolytechnique, F-91128 Palaiseau Cedex, France b Dcpartamento de Matemriticas, Estadistica y Computackin, Facultad de Ciencias, Universidad de Cantabria, E-39071 Santander, Spain c Departamento de Matemritica e Informcitica, Campus de Arrosadia, Universidad Ptiblica de Navarra, E-31006 Pamplona, Spain

Abstract We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero-dimensional polynomial equation system. This result represents a multivariate version of Liouville’s classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight-line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton’s algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras. @ 1997 Published by Elsevier Science B.V. 1991 Math.

Subj.

Class.:

68Q25,

14Q15,

11568

1. Introduction The present

paper represent

the design of algorithms

a continuation

of intrinsic

of [25, 26, 441. These

type to solve systems

papers concern

of polynomial

equations.

* Corresponding author. E-mail: [email protected].

’ Research was partially supported by the following French, Spanish and Argentinian 1026 MEDICIS,

PB93-0472~C02-02,

UBACYT-EX-001

0022-4049/97/$17.00 @ 1997 Published PII SOO22-4049(97)00015-7

and PID CONICET

3949192.

by Elsevier Science B.V. All rights reserved

grants:

GDR CNRS

278

Solving

M. Giustiet al. I Journal of Pure and Applied Algebra II 7& 1161(1997) 277-317

is then applied

By “intrinsic semantical

to decide

type” we mean algorithms

and the syntactical

both for the improvement

character

equation

of systems

of polynomial

which are able to distinguish of the input system

of the complexity

With respect to bit complexity polynomial

consistency

equations. between

the

in order to profit from

estimates.

we show how the time necessary

to solve a given

system is related to the affine degree and the (affine) logarithmic

height of the corresponding diophantine variety. In this sense, the results of [25, 261 show already that the (affine) degree of an input system is associated with the complexity when measured in terms of number of arithmetic operations. However, there are still some drawbacks

to this approach.

The first one is that the algorithms procedure and that the algorithms in the case of [25] the algorithms

developed

in [26] require iterative

calls to the

in [25] rely on the use of algebraic numbers. Thus are not “rational” although their inputs and outputs

are. A second drawback concerns the modeling tity of arithmetic operations of an algorithm

used to measure complexity. does not explain sufficiently

pens when we are using it on a “real-life” computer. models of bit complexity (Turing machines, random

Years of experience show that access machines or equivalent

models) represent more realistic patterns for practical computing. of bit complexity of the intrinsic complexity of the algorithms necessary. In the present paper we deal with both disadvantages

The quanwhat hap-

In this sense, a study of [25, 261 becomes

of the algorithms

in [25,26],

giving practicable solutions (cf. also Section 3). First, our new algorithm is completely rational. It does not require any constants other than those in the field of coefficients of the input system. Secondly, to improve the bit complexity estimates, we introduce a suitable notion of height of afline diophantine varieties which is inspired by the corresponding notion introduced shown in Section 2, our notion geometric

elimination

procedures

for projective varieties in [5,9,21,42,43,45-48]. As of height is strongly related to the bit complexity of of any kind.

Our notion of height combined with a new algorithmic interpretation of duality theory for complete intersection ideals yields a new Liouville estimate (cf. Sections 7 and 4). Liouville estimates can be applied to get lower time bounds for the numerical analysis approach to solving systems of polynomial equations. In particular, we show the practical inefficiency of both floating point and binary encoding for rational approximation of zero-dimensional multivariate polynomial equation systems even in the algorithmically well-suited cases. To illustrate these introductory observations, let us first consider the following two problems. Problem 1. Let be given a sequence of n integer polynomials with coefJicients in (0, l}, using three variables each:

of degree

at most 3,

M. Giusti et al. I Journal of Pure and Applied Algebra 117&c 118 (1997) 277-317

219

Decide whether the following system of polynomial equations has a solution: X&-X,,k=O

,..., X,$-X,,,k=O,Yf-Y,=O

,...,

1
Yjn-Y3,,=0,

n

rI i=l

Problem 2. Given in decimal representation an integer k E N and the polynomials:

XT - x, = 0,. . . ,x,’ - x, = 0, k - (Xl + 2x2+. . . +2”-‘XJ

= 0.

Decide whether this system has a solution. Both problems

have a similar

“syntactical

form” (i.e., they look very similar,

as

a consistency question for “syntactically easy” polynomial equation systems). However, they possess completely different “semantical” characters: the first problem is a translation of a well-known NP-complete problem (3_satisfiability, for short 3SAT), while the second one just concerns the binary encoding of k with n bits (if there exists such an encoding). This means that the second system is consistent if and only if k may be encoded using at most n binary digits (cf. [31, 441). Traditional symbolic procedures deal with both problems using in each case the same general treatment. However these two problems demand for different algorithms that may profit from their different semantical features. The construction of algorithms which are able to distinguish between equation systems which are semantically well suited and such which are not is the main goal of our paper. We shall also consider polynomial equation systems in the following terms:

the question

of consistency

of

Problem 3 (Effective Nullstellensatz). Given a sequence of polynomials fi , . . , fS E , . . . ,X,,], decide whether the afine algebraic variety Z[Xl V(f ,,..., fs):={XE@“:f,(X)=

..’

=fs=O}

is empty or not. As both Problems 1 and 2 above can be written as special cases of this more general Problem 3, we have to find a way to distinguish (within the context of Problem 3) between the different levels of difficulty Problems 1 and 2 represent. rithms which solve this problem taking care of the special features instance of the input system, we need two major geometric invariants: and the affine height of the system (cf. Section 2.2). Assuming these are going to show in this paper the following result.

To design algoof the particular the affine degree two notions we

280

M. Giusti et al. I Journal of Pure and Applied Algebra 117&118

(1997) 277-317

Theorem 4. There exists a bounded error probability Turing machine which solves the following task: given polynomials f,, . . . , fn+l E Z[Xl, . . .,X,1 of degree at most 2 and of (logarithmic) V( f,,

height h, the Turing machine decides whether the variety

, fn+l ) is empty or not. Moreover, iJ’ 6 is the intrinsic (affine) degree of the

system and n is the intrinsic (logarithmic) height of the system, the Turing machine answers in (bit) time

using a total amount of arithmetic operations in Q of (n6)‘(‘). Our algorithm

first solves a suitable polynomial

equation

system and then uses this

system fi,. . . , fn+l. Solving is let us assume that the polynomials fi, . . , fn form a regular sequence, each ideal (fi, . . . , fi), 1 < i 5 n, being radical. Then the algorithm proceeds in n steps, solving at each stage 1 5 i < n the system

information for the consistency test of the original done inductively. In order to explain the procedure

f, =o )...) fi=o. The corresponding intermediate algebraic varieties are obtained by a lifting process from special zero-dimensional varieties which we call the lifting jibers. The lifting process is based on a division-free symbolic version and represents a algorithmic version of the Implicit

of the Newton-Hensel algorithm Function Theorem. In this way,

the procedure constructs the lifting fiber of step i + 1 from the lifting fiber of step i. Each inductive step includes two cleaning phases: one is performed to throw away extraneous projective components (mainly those at infinity) and a second one reduces the size of the representation of the integers which appear during the process. This second cleaning

phase is included

to avoid uncontrolled

of intermediate polynomials. Finally, we show how the following division problem in the Nullstellensatz resolution

of systems of multivariate

growth of integer coefficients

two computational and the numerical

polynomial

problems are related: the analysis approach for the

equations.

Problem 5 (Division problem in the Effective Nullstellensatz). Let be given polynomials fi,. . . ,fs E Z[X~, . . . ,X,,] without any common zeroes (i.e., the polynomials satisfy

the condition

V( f,, .

, fS) = 0). Compute a non-zero integer a E Z, and poly-

nomials 91,. . . , gS E Z[X,, . . . ,X,] such that

holds. Problem 6 (Numerical analysis approach). Given a real number E > 0 and a regular sequence of polynomials fi, . . , fn E Z[X,, . . ,X,,] of degree at most d defining a zerodimensional a$fine variety V := V( f,, . . . , fn), compute for any point c(E V an

M. Giusti et al. I Journal of Pure and Applied Algebra 117& I18 (1997) 277-317

281

approximation a E @” at level E > 0, i.e., find by an eflective procedure a point a E @” such that Ila - x/I < 6 holds. At first glance Problems

5 and 6 seem to be unrelated.

However,

the solution of Problem 5 provides lower bounds for any solution shall state this observation in terms of Liouville estimates. In order to understand

the relation between

the context of Problem 6 any approximation nal (i.e., such an approximation

Problems

we shall see how of Problem

6. We

5 and 6, let us observe that in

a computer may output is necessarily

must be a point a belonging

is assumed to be encoded in binary (i.e., by the binary expansion numerators of the coordinates of a). In the past the division problem in the Nullstellensatz

ratio-

to Q[i]“). Such an output of denominators

and

was studied in both number

theory and computer science simultaneously. The first results mainly established upper bounds for the degree [lo, 13, 12, 331 and height [3,4] of the polynomials 91,. . , gS appearing in Problem 5. The complexity study in [22, 34, 351 yields optimal upper bounds for both invariants (see also [19]) and can be applied estimates in the following sense (cf. also Section 2.5). A Liouville

Liouville

estimate is a lower bound for the binary length (the logarithmic

of the numerators and denominators V, the point cx and the real number Theorem

to derive

7 gives such a Liouville

of the coordinates E in the statement

height)

of a in terms of the variety of Problem 6. The following

estimate.

Let us introduce the following notions and notations: let VcC” be a zero-dimensional diophantine variety not intersecting Q[iln and let u be a point of V. Let E > 0 be a real number.

We call a point a = (al/q, , . . . , a,/q,) E O[i]” a rational approximation

of x at

level E > 0, if

holds. (Here 11.II denotes the norm associated with the usual hermitian product in C”.) A regular sequence f, , . . . , fn E Z[&, . . . ,X,] is said to be smooth if for any 1 < i 5 n the Jacobian J( f 1,. . . , fi) is not a zero-divisor modulo the ideal (fi,. . . , f2). Observe that for a smooth regular sequence the ideals (fi , . . . , fi) are always radical. Theorem 7. There exists a universal constant C with the following property: given a smooth regular sequence of polynomials f,, . . . , fn E Z[&, . . . ,A?,,] of degree at most d and of logarithmic height at most h. Furthermore let be given a point a E Q[i]” and a real number 0 ( E 5 1. Suppose that the variety V := V(f,, . . ,fn) following conditions: - V tl Q[iln = 0. ~ there exists a point SIE V such that ((U- Xl1<

E 5

1

verifies the

282

hf. Giusti et al. I Journal of Pure and Applied Algebra I1 7& 118 (1997) 277-317

holds. Then we have for any denominator q of a the following inequality: - (h + q) 5 log, q. (ndG)C where 6 denotes the degree and ye the (afine)

Proof. According

to Proposition

E-1

logarithmic height of V.

20 below we have

q2.

lgn+l(colw4l+ 1) 5

Thus, we look for upper bounds

for IIc((J and lgn+i(C()/. First, from Proposition

14

below we deduce (21la1] + 1) < v&2”? On the other hand, combining log, lgn+i(CoI < (nd@‘(‘)(h From these two upper bounds We observe

Theorem

21 and Lemma 23 below we conclude

that

+ y + log, 4). we easily deduce the bounds

that for n = 1, Theorem

7 represents

0

of the theorem.

just a restatement

of Liouville’s

classical result with somewhat coarser bounds (cf. e.g. [50, Kapitel I, Satz 11). The height estimates in [3, 4, 19, 34, 351, combined with the methods described in Section 2.5 (Proposition 20) produce Liouville bounds that relate the syntactical description of V as given by the input, the approximation E and the height of the denominator of a. In fact, these estimates yield the inequality: - log, E < d”“‘h log, q.

(1)

On the other hand, degree and height of the solution time bounds

for the numerical

analysis

approach

set represent

to polynomial

reasonable

equation

lower

solving

by

rational point approximation. Exponential degrees and exponential height produce exponential lower time bounds for numerical methods of polynomial equation solving (cf. [44]). A lower bound like inequality (1) above, i.e., a lower bound for log, q, represents a lower bound for the output length and, therefore, a lower time bound for numerical methods of polynomial system solving. The results of [24,8,9,19,34,35,46-48] imply that an approximation level of e := 2- do’“’ is sufficient in order to characterize (and to distinguish adequately) the solutions of the variety V in Problem 5. On the other hand, the examples of [39, Chapitre 4, Proposition 121 and [29,30, Example 31 show that even in case of semantically and syntactically “easy” systems such an approximation level may be necessary. Therefore, it becomes reasonable to fix an approximation level for numerical solving of E := 2-d”, where d” represents the Btzout number of the input system. Under this assumption, lower bounds are required.

inequality

(I ) becomes

meaningless

and more precise

M. Giusti et al. IJournal of Pure and Applied Algebra 117& 118 (1997) 277-317

However

an exponential

tained in the following Corollary

lower time bound

corollary

to Theorem

for numerical

solving

283

is implicitly

con-

7.

8. Given a smooth regular sequence fi,. . . , fn E Z[X,, . . . ,_&I, and points

c(E V, a E Q[iln verifying the conditions of Theorem 7, let E > 0 be a level of approximation such that -log,

E =d”

(where d” represents

the BCzout number of the

system). Then we have d” (ndS)C

(h + YI)I log, 9,

where C is a suitable universal constant as in Theorem 7. In particular, for systems of “small” degree and height, the output length for numerical solving methods is necessarily exponential in the number of variables. For instance, polynomials:

we may consider

the following

smooth regular

sequence

of quadratic

This sequence defines a zero-dimensional variety with just two points (the degree of the variety is 2) of small height (the logarithmic height is 1). Thus, for a level of approximation E > 0 with - log, E = 2”, the lower bound obtained from inequality (1) says just: 2” -< 2O(“) log, q, whereas the lower bound level E of any solution consequence

from Corollary

8 states that every rational

of this system has exponential

of this corollary

is that both binary

bers in Q[i] are not efficient for reaching

in [52-54,56,17,18].

approximation

length.

of num-

level of approximation.

in the following

of

Thus, the main

and floating point encoding

the appropriate

alternative encoding is therefore required. Another way out of this dilemma may consist in [51] and further developed

binary

approach

Instead of approximating

An

initiated rationally

(or similarly

by floating point arithmetic) the solutions of the zero-dimensional input level E = 2Yd”, we just try to system ,fi, . . . , fn up to the appropriate approximation find approximate zeroes (in the sense of [54]) of the system, i.e., we try to find points a E Q[iln from which a suitable version of Newton’s algorithm converges quadratically to a true zero of the system.

2. Notions,

notations

and results

We are going to study the consistency question of Problem 3 (i.e., the decisional problem in the effective Nullstellensatz) only when the input system of polynomial

284

M. Giusti et al. /Journal of’ Pure and Applied Algebra 117& IICI (1997) 277-317

equations _ r
fi, . . . , fr E T&JCL,.. . ,A’,] verifies the conditions:

_ the sequence

J;,

...,fr_lis a smooth regular sequence.

These two additional an effective

version

conditions of Bertini’s

are not really restrictive. Theorem

allows

As shown in [22,24,27,34,35]

to reduce

a general

input

system

fl,..., .fs E w-l ,...,X,] to such a system. We just have to find generic Z-linear combinations

of the input polynomials

coefficients

in Z of logarithmic

such that these linear combinations

and d is an upper bound for the degrees of the input polynomials. tion of this preprocessing

we may suppose,

equations satisfy the following conditions: _ the ideals (f,, ...,fi)are radical, 1
1 the varieties

of dimension

contain

height O(n log, d), where n is the number

I$ = V(fi,.

without

Under the assump-

loss of generality,

1, . ,J;:) are complete

only

of variables

that our input

intersection

affine

n - i.

Notation 1. Given R c B an extension of a commutative ring (which converts B into a R-algebra), and an element b E B, we denote by vb : B + B the R-linear endomorphism induced by the multiplication of the elements of B by b (in the following we shall call such a linear map a homothety). If B is a free R-module of finite rank, we denote by Mb the matrix of the homothety

?Ib and by Xb E R[T] the characteristic

polynomial

of qb. Moreover, if R is a unique factorization domain, we shall denote by mb the primitive minimal polynomial of ?‘Ib. Observe that Xb and mb are manic polynomials of R[T], which respectively.

for short we will call characteristic

In order to decide whether

(fi,

compute the following items: - A linear change of coordinates

and minimal

,,...,

Y,_,+,]+O[Y,

of 6,

...,fr)represents the trivial ideal we just need to (Xl,.

. ,X,) + (Y,, . . . , Y,,) such that the following

Q-algebra homomorphism represents a Noether normalization V(,fi, , fr-, ),with the variables Yi, . . , Y+,.+l being free: R:=Q[Y

polynomial

,...,

Y,J(f

I,...,

of

the

variety

fr-,) =:B.

Observe that this means that B is an integral extension of R. _ the matrix Mf; of the homothety ~5 with respect to an appropriate R-module basis. Suppose that such a Noether normalization is available. Then the Q-algebra homomorphism R + B is injective and B is a free R-module of rank at most deg V,-, (cf. [27,28]). Under this assumption the ideal (fi, . .,fr)is trivial if and only if the matrix yf; is unimodular what means that the determinant of MI; is non-zero and belongs to Q. This comment equation systems in a very specific exactly we mean

shows how the original Problem 3 of testing consistency of polynomial can be reduced to the problem of solving polynomial equation systems geometric sense. In the next subsection we are going to explain what by this, namely geometric solving.

285

M. Giusti et al. I Journal of Pure and Applied Algebra II 7& 118 (1997) 277-317

2.1.

Geometric solving

The previous nomial

equation

considerations

puting a Noether normalization homothety

reduce the search for a consistency

system fi = 0,. . . , fr = 0, namely of the variety

~1; with respect to a suitable

Problem

test for the poly-

3, to the problem

of com-

V(J) . . . , J>_ 1) and the matrix Mb of the

R-module

basis. Assume

that Xi,. . , X, are

in Noether position with respect to the variety I’(fi, . , fr_l ), the variables being free. Then we consider the following integral ring extension: Xl,‘..,Xn-r+l already

R := C&Y, ,...,Xn-r+ll~Q~~l,...,~nll~fl,...,fr-1) Our assumptions

on fi , .

, Jr_,

imply that B is a reduced

R-module.

We are now going to explain

“geometric

solution”.

Definition

=: B.

First we need the following

9. Let R be a ring of polynomials

algebra

and a finite free

what we mean by “geometric

solving”

or

notion of primitive element of B:

over Q and R C B an integral ring exten-

sion such that B is reduced and B is a free R-module of finite rank. An element u E B is called a primitive element of the ring extension R 2 B if the degree of the minimal polynomial

m, of u equals the rank of B as free R-module,

i.e., if

deg m, = rankR B holds. Let K be the quotient field of R and B’ = K @RB be the localization of B by the non-zero elements of R. An element u E B is a primitive element of B if and only if for D:=rank,B=dimKB’ the set {l,u,...,&’ } represents a K-vector space basis of B’. The computation of the matrix Mfr is a consequence of the following “generic point” description of the K-algebra B’: this algebra is characterized by the following items (which our algorithm will compute): _ a K-vector space basis of B’; - for n - r + 2 5 i 5 n the matrices Mx, of the homotheties the given basis (these matrices

describe

the multiplication

qx, : B -+ B with respect to tensor of the K-algebra

B’ and hence also of the R-algebra B). We then obtain the matrix MfV by substituting for the variables X-,.+2,. . ,X, appearing in the polynomial f,(Xi , . . , X, ) the matrices Mx,_, i-2,. . , Mx,.(In the sequel we shall write for short MJ = f,.(M~,i_~+~,. . , Mx, ) for this substitution, interpreting ,fi as an element of the polynomial ring R[Xn_-r+~,. . ,X,,]). In this sense, geometric solving means just computing both a basis of the K-algebra B’ and the matrices M,y,_,.,, , . . , Mx~. This is done making use of a suitable primitive element of B. In the context of this paper, the primitive element u E B will always be chosen as the image in B of a generic Z-linear form of the variables &_,.+z,. ,X,. In particular, we may assume that u is the image of a linear form U = &_,.+zX~_~+Z+. . . +2,X,, with

286

hf. Giustiet al. IJournal of Pure and Applied Algebra 117& 118 (1997) 277-317

i, E L for n - r + 2 2 i < n. Let T be a new indeterminate. m,(T)

of u as an element

K-algebra

of the R-algebra

B’) will be a manic polynomial

We shall choose

this minimal

The minimal

B (or equivalently in O[Xt,. . . ,&-,.+I,

polynomial

as an element

T] = R[T]. In the sequel we shall pay special attention

Gr+l, where we have R = Z, K = Q and where m, is a polynomial

polynomial

as an element

of the

T] = Q @,zR[T]. Z[Xt,. . . ,

of the ring

to the case r = 12+ 1,

of Z[T] of positive degree

(the polynomial m, is then trivially manic over Q[T] since we may divide it by its leading coefficient, which is a non-zero integer). Discarding the content (the maximum common

divisor)

m,, by its primitive

of the coefficients counterpart

of m,, we may replace the minimal

polynomial

which we shall always denote by qu. Similarly,

case Y 5 n, we shall replace the minimal

polynomial

for the

m, E Z[&, . . . ,Xn_,.+l, T] = R[T]

by some “cleaner” (however not necessarily primitive) version qu. Finally, as { 1, u,. . ,uD-’ } is a basis of the K-vector space B’, for n - Y + 2 < i 5 n there are polynomials al” E R[T] and non-zero elements pj”) E R such that pf”‘Xi v)“‘(U) belongs to the ideal (Ji , . . , fr_ 1) in K[Xn_r+~, . ,X,1. In particular, the following identity between ideals of K[X,_,.+2,. . . ,X,1: (fl,...,

fr-1)=(4u(U),P~IU+2Xn--r+2

- $!v+2(m

..,P?‘&

we have

- $w)).

Moreover, if M denotes the companion matrix of the homothety Q, with respect to the basis { 1, U, . . . , @-I}, the matrices A~,Y_,~~,. . . , &IX, characterizing the multiplication tensor of the R-algebra B (or equivalently of the K-algebra B’) are given by the formula Mx

=

’

for n - r+2

(p!“’>-’ .p(M) I

I

5 i 5 n. To simplify notations

we shall often omit the superindex

(u) when

referring to these polynomials. After these explanations we shall assume, without loss of generality, that we have s := IZ+ 1 in the statement of Problems 3 and 5. Consequently, we restrict the meaning polynomials

of “geometric

solving” to the case where we have given as input

Ji , . . . , fn E Z[& , . . ,X,] forming a regular sequence

in Q[Xt,. . .,X,1 (we

shall say in the future for short that ft, . . . , fnEZ[X,,. . ,X,J is a regular With these conventions “geometric solving” means the following: Definition

10. An algorithm

for geometric

sequence).

solving is a procedure which from a smooth

regular sequence f,, . . , fn Ei&Y,,.. . ,X,J as input produces: _ a primitive element u of the ring extension Q -_[x,,...,x,]/(fi,..., fn) represented by a Z-linear form U = I,&, + . . . +i,X,, - the primitive minimal polynomial qu E Z[T], - the parametrizations of the variety V(fr,. . , fn) by the zeroes of qu, namely the (unique) primitive polynomials p(,u’Xt -u:“‘(T), . . . , p$f’X, -v?)(T) with py), . . . , pp’ non-zero integers and vi”‘, . . , up’ E Z[T] which satisfy the conditions max {deg$),...,degv?)}
287

M. Giusti et al. I Journal of Pure and Applied Algebra I17& 118 (1997) 277-317

In the sequel we shall refer to the polynomials (with

image

u in Q[_Xl,...,x,l/(f~,...,f~>>,

(u) PI , . . , p?’ E Z

U = Al&+

consider

we cannot

system f, = 0,.

solving”

..,fn= 0.

has a long history, which

give here in full detail for mere lack of space. One might

[ 15,231 as early references

time in complexity

E Z[X,, . . ,X,]

qu(U, v(,U’(T),...,~~‘(T)EZ[TIand

as geometric solution of the equation

Let us remark here that this notion of “geometric unfortunately

. . . +A,&

where this notion was implicitly

used for the first

theory.

2.2. Intrinsic parameters We said in the introduction that we are interested ing which are able to profit from “good” geometrical of polynomial

equations

might possess,

in algorithms for geometric solvproperties which an input system

This makes it necessary

to precise what such

“good” geometrical properties may be and how to find measures for them. Therefore, we are going to define two geometric invariants in this subsection that will arise as parameters arithmic

of the complexity

height of complete

of our procedures: intersection

the (affine) degree and the (affine) log-

varieties.

The notion of degree has been taken

directly from [28], while our notion of height is strongly inspired by the corresponding notion developed for projective varieties in [42,43,4548]. Let us first recall the notion

of degree of an affine algebraic

defining the degree of a zero-dimensional positive-dimensional complete intersection To fix notations,

variety.

We begin by

variety and then we extend this notion algebraic varieties.

let V C C” be an algebraic

subset (variety)

to

given as the set of com-

mon zeroes in C=”of a smooth regular sequence of polynomials f,, . . . , fi E i&Y,, . . ,X,] of degree at most d. If V is zero dimensional (i.e., if i = n) the degree of V is defined as the number of points of V (points at infinity are not counted in this definition). In the general

case (when dim V = n - i > 1 holds)

affine linear subspaces

of C” of dimension

i (defined

let us consider

the class 9 of all

as the set of solutions

in C” of

a linear equation system Li = 0,. . . ,L,_i = 0 where Lk = aklX,+. . +ah,X, + ako is an affine linear polynomial with coefficients akj E Z for 1 5 k
and assumptions

be as before. The degree of V is defined

of the degrees of the intersections of V with affine linear We denote the degree of V by deg V.

spaces

As observed in [28], this definition of the degree never yields infinity, but gives always a natural number. Our definition is equivalent to the following one: Consider all linear changes of coordinates in C” defined by non-singular matrices with integer entries, i.e., linear changes of the type

(Xl,...,&)

H

(YI>...,Y,),

288

M. Giusii et al. IJournal of‘ Pure and Applied Algebra 117& 118 (1997) 277-317

. +ak,X,

where Yk= ~1x1~.

is a linear form with integer coefficients

Any generic linear change of coordinates

induces an integral ring extension

~o[~l,...,r,-il-o~~I,...,ml/(fl,...,fi)

This rank is the same for any generic

equals the degree of the variety

as follows:

= Q[Vl.

It turns out that the ring O[V] is a free Q[Y,, . . . , Y,_i]-module for instance).

for 1
of finite rank (cf. [27]

linear coordinate

change

and it

V.

In order to define the height of an affine diophantine the zero-dimensional case and then the case of positive

variety, we also consider dimension.

To start with let us first say what we mean by the (logarithmic)

first

height of an integer,

a vector of integers, a matrix over Z and a polynomial with integer coefficients. Let a EL be an integer, then the height of a is defined as ht(a) := max{log, Ial, 1). It is obvious

that the height measures

the bit length of a. On the other hand, we shall

see soon that this simple notion of height of an integer has a natural extension to algebraic varieties where it plays the role of “arithmetic degree” (see [9, 5, 21, 34, 35, 46-481). The main outcome of this paper will be the reinterpretation of the notion of height for algebraic varieties as a measure for the bit complexity of an elimination procedure. In this sense, our contribution justifies a posteriori Northcott’s terminology of “complexity” for the height of an algebraic variety [57]. For a vector of integer numbers ~:=(a,, . . , a, ) E Z”, we define the height ht (a) as the maximum of the heights of its coordinates. For a matrix A E J&(Z) with integer entries, its height ht (A) is defined as the height of A as vector. Similarly, we define the height of a polynomial Definition

12. Given

f E Z[Xl,

. .,X,,] as the height of the vector of its coefficients.

a zero-dimensional

diophantine

algebraic

variety

V G C” and

a linear form U = i&l+. . . +A,& with integer coefficients representing a primitive element u of the ring extension Q + Q[V], we define the height of V with respect to U as the maximum X, - v?‘(T)

of the heights of the polynomials

(see Definition

In our first approach

qu(T),py)&

- u{~)(T),

. . . ,pt’

10) and denote this height by ht ( V; U).

to find a suitable

notion

define the height of the given zero-dimensional

of height for algebraic variety

V as the function

varieties

we

ht, : N + N

which associates to any natural number c E RJ the value htv(c) := max{ht (V; u): ht(U)Lc} if the ring extension Q + O[V] has a primitive element of height at most c and which associates to c the value 1 if no such primitive element exists. This notion of height is related to the hermitian norm and the denominator of the points of the variety V. In order to explain this relation, let us introduce the following notations. Taking into account that the zero-dimensional variety V is contained in C”, we define the norm [[VII as

11 VII :=max{llccI1 : a

E

V}.

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M. Giusti et al. I Journal of Pure and Applied Algebra 117& 118 (1997) 277-317

Furthermore,

a natural

number

d E N is called a denominator of V if all elements

in

the set

d.V:={d,a:rE have algebraic

V}

integers

the denominator

as coordinates.

The smallest

denominator

of V will be called

of V and denoted by dv.

With these notations dimensional varieties.

we are able to state the following

height estimation

for zero-

Lemma 13. There exists a universal constant K > 0 such that for any zero-dimen-

sional diophantine subvariety V of @” and any c E N the following inequality holds:

htdc) 5 WC + log,(ndv IIVII1). (This means the function htv is of order htv(c) = 6’(‘)(c + log2(ndvI( V(l))). Proof. Let V c @" be a zero-dimensional

diophantine

variety

of degree

6 E N. The

inequality is trivial for htY(c) = 1. Therefore, we may suppose, without loss of generality, that the ring extension Q --f Q[ V] has a primitive element u which is the image of a linear form U=2,& +... + ,I,& E Z[Xl, . . ,X,] of height at most c. We just show the inequality:

and leave the corresponding

inequalities

for py’&

- vl”‘( r), . . . , pfiu?y, - v?‘( 2’) to the

reader. Since dv is a denominator for V and U has integer coordinates, dv is also a denominator for the set of algebraic numbers u(V) contained in C. Therefore, if OL,,.

, CQE C” are the points of V and if 7’ is a new indeterminate,

f(T):=fI(T

the polynomial

- dvu(ai))

i=l

has integer coefficients,

i.e., f(T)

belongs

to H[T]. Moreover,

f (dyT)

the polynomial

vanishes on the set u(V). Since f and q,, have the same degree 6 and since qu is the primitive minimal polynomial of the image u of U in Q[V], the vanishing off (dyT) on u(V) implies the existence of a non-zero integer b E Z such that f(T) = bqu(T) holds. Taking into account Ib(2 1 and that the coefficients of f are elementary symmetric functions in the points of the set dv . u(V) c @ we deduce the following inequalities:

2ht(~~)<2hr(/)<(2dv(lu(V)ll)6<(2d~)6(n2C)6((V(ld<(2C+‘ndvI/VII)6. Again, let V be a zero-dimensional diophantine estimate now the quantities dv and 1)V/j.

subvariety

0

of C” of degree 6. We

290

M. Giusti et al. IJournal of‘ Pure and Applied Algebra 117&118

Proposition

(1997) 277-317

14. Let c E N be a natural number such that the ring extension Q +

Q[V] has a primitive element u := ,?,A!, + . . + ;I,,& E Z[X,, . . .,X,1 of height at most c. Then dr and (/V 1) can be estimated as follows:

(i) where a~ is the leading coefficient of q,(T) and nb, pi(‘I is the discriminant obtained from the polynomials in Definition 10. It follows that V has denominators of height O((n + Wv(c)). (ii) the norm 11 V /( of V satisjes 11V]j < +Y26h’” Cc). Proof. The integer la&l is a denominator of the set u(V) because it is the leading coefficient of the polynomial qu(T) which defines u(V). Since the polynomials py) , . . . , p?‘X, -v?)(T) “parametrize” the variety V in function of the zeroes of qU(T), i.e., in function of the set u(V), we deduce that la:-’ ny=, p$‘)/ is a denominator

X,-$)(T)

of V (notice that the degree of the polynomials From [39, Chapitre IV, Theo&me

U?‘(T), . . . , up’(T)

is bounded by 6- 1).

2 (ii)], one deduces

liU( V)]l 5 2ht’,(c).

V and l
Thus, for u=(cI,,...,R~)E

we have

IQ] = Ip~‘I-‘~c’“‘(u@))l. k Since pr’

is an integer

and ok(T) is a polynomial

of degree at most 6 - 1 and of

standard height at most 2hty(C), we conclude IOLk 15 62h” Cc)\\U( v)((“-’

5S26htc

This implies that

[[VI]:=max{ I/MI]: a E V} < fiLi2”h”(C). Remark 1. The height is strongly

related with complexity

issues in polynomial

equa-

tion solving. In fact, given a regular sequence of polynomials fi , . . . , fn E E[Xl, . . . ,X,] defining a zero-dimensional diophantine variety V, we have an upper bound for the length of the output of any algorithm which solves the system fi = 0,. . . , fn = 0 geometrically, namely: (n + 1) deg V htY(c). Here we assume tacitly that the output is given by a linear form representing a primitive element of the ring extension Q + Q[ V] and by polynomials as in Definition 10 and that these polynomials are given in dense and their coefficients in bit representation.

292

M. Giusti et al. IJournal of’ Pure and Applied Algebra 117&118

In a more down-to-earth 6 = ~‘(a)

language

to be smooth

points

this means

(Observe & implies notations

that everything

of

deg r;; of elements

of I;; to

. , Y&i]-module

Q[ V]. Furthermore,

we ask

a primitive

comes

that we ask all the elements

of V and the number

be equal to the rank of the free Q[Yl,. the linear form U to generate

(1997) 277-317

together

that the zero-dimensional

element

of the ring extension

since the smoothness

Q-algebra

of the elements

O[&] is reduced.)

we model our notion of height of algebraic

Q + Q[ K].

Maintaining

of

these

varieties by means of the follow-

ing function. Definition

17. Given

V a complete

intersection

diophantine

variety as before, a linear

change of coordinates AE .Nv, a linear form U E Z[Yn_i+l,. . . , G] whose image u in Q[ V] is a primitive element of the integral ring extension (2) and given a point a E Z& we define the height of V with respect to the triple (A, U,a) as ht(V;(A,U,a)):=ht(&U). Our first approximation

to the notion of height of a diophantine

complete intersection

variety is given by the function htv : N -+ N which associates to any natural number c E N the value htv(c) := max{ht(V; (A, U,a)) (ht(A, U,a)
programs subsections

we discussed

the mathematical

form and syntactical

en-

coding of the output of the algorithm for geometric solving we are going to exhibit. However, we also need a suitable encoding for input and intermediate results of our algorithm. As mathematical objects our inputs are polynomials with integer coefficients f E Z[X,,

. ,X,1. These polynomials

yields the first two possible sparse encoding.

can be written

encodings

Thus, if d is the degree of f

height, the length of f under dense encoding

Let us remark that the binomial

as lists of monomials

for input and intermediate

coefficients

results:

and this dense and

and if h is an upper bound

for its

is

appearing

in this expression

are poly-

nomial in the number of variables in the case of “small” degree polynomials (e.g., the length of the dense encoding of f is h. (“l*) z h. n* if the degree of f is at most 2). Analogously, the length of the dense encoding of f is polynomial in d for d + n. Alternatively, our input polynomial f may be given by representing just all its nonzero coefficients: this is the sparse encoding of the polynomial f.Then, if h is a bound of the height, d the degree and N the number of monomials with non-zero coefficients of f, the length of the sparse encoding of f becomes hNn log, d.

M. Giusti et al. I Journal of Pure and Applied Algebra 117&118

However, programs. version

in many

practical

applications

This is, for instance,

of the 3SAT problem

in the following Definition

the approach

(Problem

we shall mainly restrict ourselves

our input

in the encoding

to the straight-line

may be given

of the inputs

program encoding

as

of the

In the sequel of polynomials

sense.

18. A generic straight-line of the graph. 9 contains

the variables

polynomials

293

277-317

1) we stated in the introduction.

program

r’ over Z is a pair r’ = (99, Q), where

$9 is a directed acyclic graph and Q is an assignment vertices)

(1997)

Xl,.

of instructions

it + 1 gates of indegree

. , X, and by the constant

to the gates (i.e.,

0 which are Q-labeled

by

1 E Z. They are called the input gates of

r’. We define the depth of a gate v of the graph 9 as the length of the longest path joining

v and some input gate. Let us denote the gates of the directed acyclic graph 9’

by pairs of integer numbers corresponding

(i,j),

value of an arbitrary

where i represents numbering

(this encoding can be seen in [26,34,40,41,44]). (i,j) we have the following operation:

where Ayj,B$’

are indeterminates

puted values corresponding

the depth of the gate and j is the

imposed

on the set of gates of depth i

Associated

called parameters

to the intermediate

gate

of r’ and Q,.s,Qr/sl are precom-

to the gates (Y,s) and (Y’,s’).

We denote by 2 = (Ay), B = (B$“) the list of all parameters in the straight-line program r’. The intermediate results Q;j of r’ are therefore polynomials belonging to Z[k,B,X, , . . . ,X,] and r’ represents a procedure which evaluates them. A (finite) set of polynomials

fi, . . . , J1 E Z[X,, . . . , X,] is said to be evaluated

by a straight-line

pro-

gram of generic type r’ with parameters in a set 9 C Z if specializing the coordinates of the parameters k and B in r’ to values in 9, there exist gates (il,jl), . . . , (&,j,) of r’ such that

holds for 15 k 5s. Specializing in the indicated way the parameters of r’ into values of 8 we obtain a copy r of the directed acyclic graph 9 underlying the generic straight-line program r’ and of its instruction assignment Q. We call this copy r a straight-line program (of generic type P) in Z[X,, . ,X,,] with parameters in F. The gates of r correspond to polynomials belonging to Z[X,,. . . ,X,1. These polynomials are obtained from the intermediate results Qij of r’ by specializing the parameters on which they depend. We shall call these polynomials the ate results of the straight-line program I-. Furthermore, the polynomials fi, called (final) results or outputs of r. Alternatively, we shall say that fi, represented, computed or evaluated by r

adequately intermedi. . . , fs are . , fs are

294

A4. Giusti et al. IJournal of’ Pure and Applied Algebra 117~6 118 (1997) 277-317

The current complexity measures for the generic straight-line - the size of r’= the size of the graph 3, - the non-scalar depth (r’) = the depth of the graph 3. The size and non-scalar gously

defined.

parameters

depth of the specialized

Additionally,

for any straight-line

in F G Z, we call the maximum

height of the parameters nomial f E Z[Xt, .

straight-line program

r

program

r’ are:

program

r are analo-

in Z[Xt,. . . ,X,]

of the heights of the elements

of r or, for short, the height of r. The encoding

,X,] by a straight-line

in 9

with the

of a poly-

program r in Z[Xt, . . . ,X,] with parameters

of height h has (bit) length 2(L2h + L log, L). Note that the notion of straight-line program encoding covers as well the both notions of dense and sparse encoding of polynomials. This can be seen as follows: a polynomial of degree at most d and height at most h, given in dense encoding,

can be evaluated

by

a straight-line program of size O(d (“r)) an d non-scalar depth O(log, d) with parameters of height at most h. Similarly, if a polynomial f E Z[X,, . . . ,X,] has degree at most d, height h and N non-zero program

of size O(dN)

coefficients,

and non-scalar

it can be evaluated

by a straight-line

depth O(log, d) with parameters

of height at

most h. 2.4.

Complexity

of geometric

solving

Now that the notions of geometric

solving, degree and height of polynomial

equation

systems and straight-line programs and its distinct complexity measures have been introduced, we are able to state our main result, namely the following Theorem 19. This theorem represents our principal contribution to the solution of Problem 3 in Section 1.

Theorem 19. There exists a bounded error probabilistic

Turing machine

that from

a smooth regular sequence fi,. . , fn E Z[Xl,. .,X,1 outputs a linear form UER [XI,. . . ,X,,] representing a primitive element u of the ring extension Q + Q[ V], where V := V( fi , . . , fn) is the algebraic variety defined by f,, . . . , fn, and polynomials of the form q,,(T), p\“‘_& - v~“‘(T), . . , p??u, - vr)( T) with pi’), . . . , pr’ non-zero integers (u) “I E Z[T] such that these polynomials represent a geometric solution and qU,v1 , . . , on of the equation system fi = 0,. . . , fn = 0 (see Definition 10). The Turing machine finds this solution in time (counting the number of bit operations executed) (ndhLog)O(‘%> n+‘) using only (ndLG)‘(‘) arithmetic operations in Z at unit cost. We assume that the polynomials have degree at most d and that they are given by a straight-line program

f,, . . , fn of size L

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M. Giusti et al. 1Journal of Pure and Applied Algebra lI7& 118 (1997) 277-317

and non-scalur depth 8 with parameters of height at most h. The quantity 6 dejined by 6:=max{deg V(f, ,..., fi): 1 liln} is the degree of the system f~ =0 ,..., fn =O. Finally, the quantity n is the height of the system f, = 0,. . , fn = 0 defined as y := max{htv,(cs((log, n + a) log, 6)): 15 i 0 is a suitable universul constant independent of the specijk input J;, . . . , fn (or even its size). Theorem 19 follows from the description of the algorithm given in Section 3. Theorem 4 follows immediately from Theorem 19 by putting d := 2, L := n2, e := 2. Let us remark here that Theorem

19 improves

and extends the main result of [25,26]

to the bit complexity model. It sheds also new light on the main complexity of the papers [54,20, 111.

outcome

2.5. A division step in the Nullstellensatz In this subsection

we show how Nullstellensatz

bounds

imply Liouville

estimates.

This establishes a close connection between Problems 5 and 6 in the Introduction. us recall the assumptions in the statement of Theorem 7.

Let

There is given a smooth regular sequence of polynomials f,, . . . , fn E Z[X,, . . . ,X,J of degree at most d and of height at most h, defining a zero-dimensional affine variety V:=V(fi,...,f,)C@“.Moreover,thereisgivenapointa=(ccl,...,cc,)EVandareal number 0
of Theorem

7. For this purpose consider

integers

such that p/q is an approximation at level E to the algebraic We introduce the following polynomial:

p E Z[i] and q E IkJ

number

~(1.

fn+l :=(q& - p)(q& - j) of p). The assumption V n (Q[i] x C-’ ) = 0 f,, . . . , fn+l has no common zero in C”. Thereinteger b E Z and polynomials gi, . . . , gn+i E Z[Xi, . . . ,&I

(here p stands for the complex conjugate implies that the sequence of polynomials fore, there exists a non-zero such that the following B&out

b = ah Evaluating

identity

holds:

+ . . . + gn+lfn+l

this identity

at the point CIE V, we obtain

1I Ibl = lgn+l(~)l+m This yields the following

PIN

-

iii.

estimate.

20. With assumptions and notations as before, for any rational approximation a := (PI/q,. . , p,,/q) E Q[i]” at level E to the point IXE V we have the

Proposition

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M. Giusti et al. I Journal of’ Pure and Applied Algebra 117& I18 (1997) 277-317

inequality:

In particular, for [Ia - ~111
E-1

Ic7n+~(~>l(21141 + l>“** (Here pl,. . , p,, E Z[i] are Gaussian integers and q E N is a natural number.) From the second inequality we deduce that it is sufficient to bound the values of llclll and jgn+t (a)/ in order to obtain a Liouville estimate for the rational approximation a of the point CI.Note that the bounds for llcl]l and jgn+r(a)l which can easily be deduced from the known Nullstellensltze (as, e.g., in [3,4, 341) would be insufficient for our purpose (proving

Theorem 7) as they imply only a unspecific

- log, E dc”h < log,

general estimate, namely,

I41

for a suitable constant C > 0. A Liouville estimate of this type does not take into account the specific properties of the variety V expressed through its degree and height. The more specific bounds rem 7 by means result:

for I/M]/ and [g,,+,(a)] as required

of Proposition

20 are an immediate

for the proof of Theo-

consequence

of the following

Theorem 21. Let f,,

, fn, fn+, E Z[X,, . . . ,X,,] polynomials having no common zero in C” and vertfying the following assumptions: _ there exists a straight-line program of size L and non-scalar depth e with parameters of height h that evaluates the polynomials fi, . . . , fn, f,,+l; _ the degrees of the polynomials fi, . . , fn+l are bounded by d and h’ is an upper bound for their heights. Let us furthermore assume that f,,. . ., fn form a smooth regular sequence which

dejines a zero-dimensional afine algebraic variety V = V( fi, . . . , fn ). We also consider the following quantities: ~ o:=deg(V); - v]:= min ht (V) (i.e., n is the minimal value of htv distinct from 1). Then there exists a straight-line program of size L(ndG)‘(‘) and non-scalar depth of order O(log, n + log, d + log, 6 + 8) with parameters of height at most O(max{h, h’, n, log, n, 8)) which evaluates a non-zero integer a E Z and a polynomial gn+l E i&Y,, .,&I such that

a-

gn+l . fn+l

belongs to the ideal (fi,. . ,, fn) generated by fi,. . ., f,, in E[X,,. . .,X,].

M. Giusti et al. I Journal of Pure and Applied Algebra 1178~ 118 (1997) 277-317

The proof of this theorem in Section 4. Applying bound

for the value

will follow from the description

Theorem

21 and Lemma

Jgn+i(C()]. From this bound

we then deduce easily Theorem

of the algorithm

23, we obtain together

297

given

a more precise upper

with Proposition

14 and 20

7.

3. An algorithm for geometric solving The aim of this section is to establish a proof for Theorem 19. We describe an algorithm which implies Theorem 19. This algorithm works inductively on the codimension of the varieties recursion.

6 := V( ,f,, . . . , fi),

1 5 i < n, and our main goal is to describe this

Recall that our input is a smooth regular sequence fi, . . , fn E Z[& , . . . ,X,] of degree at most d. We assume that this input is encoded by a straight-line program r of size L and non-scalar depth C with parameters of height at most h that evaluates the polynomials ,f; . . . , fn. Our algorithm computes a geometric solution of the zero-dimensional algebraic variety V := V( fi, ..,fn ) C C".In order to describe for 1 < i < n the ith recursive

step of our algorithm,

algebraic varieties - 6, := deg( K), ~ 6:=max{6;: _

we shall refer to the intermediate

as I$ := V( f,, . . . , f;)and introduce

complete

the following

intersection

parameters:

1
q, := ht,( C), where C is a suitably chosen natural number of order O((log, n + t’) log, 6) such that Q[K] has a primitive element of height C with respect to a suitable

Noether position

of I$.

- q:=max{y,: 1 . “n Y(‘))

we shall obtain geometric computes

for every

solutions

of the algebraic

1 5 i < n a Z-linear

change of

>

such that the ring extension R; :=

is integral.

O[r,“‘,..., &y’,l-+B; The Rj-module

:=

Q[y,(‘),. . . ) r,c”ll(fi,...,fi>=a[~l

Bi is a free module

of rank D, 5 hi. Let us denote

xi : r/;+ Cnpi the projection on the first n - i coordinates phism TI, of affine varieties is finite.

of (I’,(‘), . . , gi’).

by

The mor-

Definition 22. Let assumptions and notations be as before. A lifting point for W := I/; of the finite morphism ri is a point P = (PI,. . . , pn-, ) E Zn-i with the following properties: _ the zero-dimensional fiber Wp := n;‘(P) has degree (i.e., cardinality) equal to the rank of B, as free &-module (this means deg( Wp) = Di)

298

M. Giusti et al. IJournal of Pure and Applied Algebra 117&118

_ the fiber W, contains

only smooth points. (This is equivalent

to saying that the Jaco-

we obtain from fi ( Yin), . . . , K?), . . . , fi( q@), . , IL ’ )

bian matrix of the polynomials by substituting

(1997) 277-317

for Yi’ ,. . ., l$)i the coordinates

p1 ,...,p

,,_ I 0fP

is ’ regular at evk:,

point of the fiber WC’; P. If P E TFi

is a lifting point of the morphism

of W = K. Observe that the elements

rci we will call its fiber Wp a lifting $ber

of a lifting fiber of rti are smooth points of W.

The lifting fibers Wp have the property that a geometric be reconstructed

from the projection

solution of the variety K can

xi and any geometric

such a fiber (see Section 3.2). Our algorithm lifting point I”r:E Zflhi of the morphism

solution

3.2, treating

how it is possible

“lifting

to reconstruct

of

will choose for each 1 < i 5 n a suitable

ni. In the sequel we shall denote the lifting

fiber of this point by Vc. The following Section 3 is divided into two well-distinguished (i) Section

of the equations

by a symbolic

a geometric

Newton

solution

parts, namely:

method”,

of the equations

where we show fi, . . , fi from

the lifting fiber Ve. (ii) Section 3.3, where we show how to find a linear coordinate change (Xi,. .., X,) (1), . . . , I$“), the lifting point P and a geometric solution of the equations of the -
of V,

by the quantity

Ci+ l)(si + 2)?/i. 3.1. Elementary

operations

and bounds for straight-line

programs

In this subsection we collect some elementary facts about straight-line programs. We start with an estimate for the degree and height of a polynomial given by a straightline program. In order to state our result with sufficient generality, let us observe that the notion of height makes sense mutatis mutandis for polynomials over any domain equipped with an absolute value. Lemma 23 (Krick and Pardo [34]). Let R be a ring equipped with an absolute 1.1: R--t 53. Suppose f E R[Xl, . . ,X,] is a polynomial which can be evaluated

value by a

M. Giusti et al. IJournal of Pure and Applied Algebra 1178~ 118 (1997) 277-317

299

straight-line program T in R[&, . . , , X,,] of size L and non-scalar depth & with parameters of height h. Let H > 0 be a real number and let u = (ccl,. . . , a,,) be a point in KY’such that log, ]uil 5 H holds for 1 5 i 5 n. Then f and f(a)

satisfy the following

estimations: - d&f) ~ ht(f)

5 2”, 5 (2r+1 - l).(h+log,L), - log, IS(a)1 5 (2/” - 1). (max{h,H}

+ log, L).

One of the main ingredients used in our procedure below is the efficient computation of the coefficients of the characteristic polynomial of a matrix. We shall use for this task Berkowitz’ historical

division-free

predecessors

and well parallelizable

of this algorithm).

algorithm

This is the content

[6] (cf. also [ 16,7] for of the next lemma.

Lemma 24 (Berkowitz [6] and Krick and Pardo [34]). Let R be a domain. There exists a straight-line program of size No(‘) and non-scalar depth O(log, N) with parameters in { - 1, 0, 1} that from the entries of any N x N input matrix over R computes all coeficients

of the characteristic polynomial of the given matrix.

We use this algorithm

not only for the computation

of the characteristic

of a given matrix but also for the computation

of the greatest common

given univariate

in a unique

polynomials

with coefficients

factorization

polynomial

divisor of two domain

(this

task can be reduced to solving a suitable linear equation system corresponding to the Bezout identity over the ground domain. See [34] for details). The following result is an immediate consequence of the formal rules of derivation. Lemma 25. Let R be a domain. For a given a finite set of polynomials fi, . . , ,x7 of R[X,, . . ,X,,] which can be evaluated by a straight-line program p in R[Xl, . . . ,X,,] of size L and non-scalar depth t?, there exists a straight-line program in R[X,,.. .,X,,] of size (2n + 1)L and non-scalar depth e + 1 with the same parameters as p which evaluates f,, . . . , fS and all the first partial derivatives:

Combining this lemma with Lemma 24, one concludes: let f,, . . . , fn be a family of polynomials of R[X, , . . . , X,] which can be evaluated by a straight-line program fl of size L and non-scalar depth e. Then there exists a straight-line program in R[Xl, . . . ,X,,] of size n’(‘)L and non-scalar depth O(~@+log, n) with the same parameters as p which evaluates the Jacobian determinant:

J(fl,..‘,

af;

( >

fn):=det ax,

,<,jln’ -2

In some exceptional cases the straight-line programs we are going to consider might contain divisions as operations. Since we are only interested in division-free straight-line

300

M. Giusti et al. I Journal of Pure and Applied Algebra II 7& 118 (1997) 277-317

programs,

the following

[58] becomes Proposition

“Vermeidung

von Divisionen”

technique

due to V. Strassen

crucial. 26 (Krick and Pardo [34] and Strassen

[SS]). Let P be a (division-free)

struight-line program in Z[X,, . ,X,] of size L und depth G with parameters of height h that computes a finite set of polynomials fb, . . , J;n of Z[X,, . . .,X,,]. Assume that fo # 0 holds and that fo divides f; in Z[X, , . . . ,X,J jbr any 1 < i 5 m. Then there exists a (again division-free) straight-line program P’ in Z[X,, . . . ,X,] lowing properties:

with the fol-

(i) r’ computes polynomials PI,. . . , Pm of Z[X, , . . . ,X,,] and a non-zero integer 8 such that for any 1 < i <: m holds

p.&C

fo.

(ii) r’ has size of order 0(22’(L + n + 2’ + m)), depth of order O(L) and its parameters have height of order max{h,O(l)}. Moreover, the height of 6 is of order: 2’t’)(max{h,

e} + log, L).

The proof of this proposition is based on the computation of the homogeneous components of a polynomial given by a straight-line program. This proof also provides an algorithm computing the homogenization of a polynomial given by a straight-line program.

This is the content

of the next lemma.

Lemma 27 (Krick and Pardo [34]). Suppose that we are given a polynomial P:= CPpXF’ . ..XF in Z[Xl , . . . ,X,,] which can be evaluated by a straight-line program r of size L and depth e with parameters in a given set 9 c Z. Let be given a natural number D. Then there exists a straight-line program T’ in Z[X,, . . . ,X,,] which

computes all the homogeneous components of P having the following properties: S uses parameters from 9, has size (D + l)*L and non-scalar depth 2d. In Section 4 we shall work with a specific polynomial which we call the pseudoJacobian determinant of a given regular sequence. We introduce now this polynomial and say how it can be evaluated. Let R be a domain containing Q. Let K be the field of fractions of R and let f,, . . . , fn E R[X, , . . . ,X,J be a regular sequence in K[&, . . . ,X,1. Furthermore, let I’j, . . . , K be new variables. We write Y=(K,...,Y,). Fix l
301

M. Giusti et al. I Journal of’ Pure and Applied Algebra 117& 118 (1997) 277-317

with

lk,j

R[ 8,. . . , Y,,X,,

E

A =(lk,jh
. . , X,,]. Let us consider

the determinant

A of the matrix

namely,

A := det(A). This determinant nomials

is called the pseudo-Jacobian determinant of the regular sequence

f, , . . , fn. If d is a bound

polynomials

are given by a straight-line

for the degrees

program

of

of fi , . . . , fn and these poly-

/I of size L and non-scalar

depth P, then

there is a straight-line program /I’ of size (nd) O(‘)L and non-scalar depth O(log, n + e) which evaluates the pseudo-Jacobian determinant A. The straight-line program p’ uses apart from the same parameters as /I only parameters of Z of height O(log, d). We shall also consider the execution of straight-line programs in matrix rings. The situations

where we apply these considerations

will be of the following

type: Let R be

a domain. Suppose that there is given a polynomial g E R[Xl, . . . ,X,] by a straight-line program ,5 in R[X, , . . . ,X,] of size L and non-scalar depth e. Suppose also that there are given n commuting D x D matrices Ml,. . . , M, over R. In such a situation the entries of the matrix y(Mi , . . . ,M,,) can be computed from the entries of Ml,. . . ,M,, and the parameters of fl by a straight-line program /I’ in R of size D’(‘)L and non-scalar depth O(Y). The new parameters of /I’ are just the values 0,l (see [27] for details). Another important aspect of our main algorithm is its probabilistic (or alternatively its non-uniform) character. This is the content of the next definition and proposition.

Definition 28. Let be given a set of polynomials

YY c Z[Xi, . . . ,X,1. A finite set Q C Z”

is called a correct test sequence (or questor set) for w belonging to w the following implication holds:

if for any polynomial

J

f(x) = 0 for all x E Q implies f = 0. of Z[Xi,. . . ,X,]

Denote

by “Ilr(n,L,e),

the class of all polynomials

evaluated

by straight-line

programs of size at most L and of non-scalar

The following of moderate

result says that for the class W(n,L,&)

which can be depth at most e.

exist many correct test sequences

length.

Proposition 29 (Heintz and Schnorr [32] and Krick and Pardo [34]). Let be natural numbers n, e, L with L 2 n + 1 and consider the following quantities: u := (2/+’ - 2)(2’ + l)*

and

given

t := 6 (eL)2.

correct test seThen the jinite set { 1,. . . , u},’ c Z”’ contains at least u”’(1 - u+) quences of length t for W(n, L, e). In particular, the set of correct test sequences for W(n, L, /) of length t containing only test points from { 1,. . , u}~ is not empty. From Lemma 23 we deduce the following complexity estimate: Let f E Z[XI , . . .,X,] be a polynomial given by a straight-line program of size L and non-scalar depth e with parameters of height h. Let c1E Z” be a point of height h’

302

M. Giusti et al. I Journal qf’ Pure and Applied Algebra 117& I18 (1997) 277-317

given in bit representation. which computes

the bit representation

In the next subsection Newton iteration.

Then there exists a (deterministic) of the value f(a)

ordinary Turing machine

in time (2e~max{h,h’})o(1).

we shall make use of a problem adapted version of the Hensel-

We are now going to describe a suitable division-free

symbolic

form

of this procedure. Let R be a polynomial

ring over Q, let K be its field of fractions and let fi, . . . , fn E

R WI , . . . ,X,] be polynomials a (division-free) straight-line assume that the Jacobian nomials

of degree at most d. Suppose that fi, . . . , fn are given by program /zI of size L and non-scalar depth 8. Let us also

matrix D(f)

:=D(fi,.

. . ,fn) := (dfi/i3Xj),,,,jl,

fi , . . . , fn is regular. We consider now the following

Nf(X,,...,X,):=

(1)

-D(f)-’

The

next

($ ,..., $) lemma

gives

operator:

(3)

functions of K(Xl,. . . ,X,). This is which we denote by Nj. For any

k E N there exist numerators (#‘, . . . , g$‘) E R[X, , . . ,X,J and a non-zero /z(‘) E R[&, . . . ,X,J such that Nj can be written as

N;=

of the poly-

.

(;;;Y;::)

This operator is given as a vector of n rational also true for the kth iteration of this operator,

Newton-Hensel

denominator

EK(X ,,..., X,,)“.

a description

R[&, . . ,X,,] that evaluates numerators

of a division-free and denominators

straight-line

program

in

for Nj.

Lemma 30. Let notations and assumptions be as before. Let k be a natural number. There exists a straight-line program in R[X, , . . . ,X,,] of size O(kd’n’L) and non-scalar

depth O((log, n + Qk) with the same parameters as p which evaluates numerators WI (k) 91 Y...,Sil and a (non-zero) denominator hck) for the k-fold iteration Nf” of the Newton-Hensel operator Nf. Proof. Let A(f) = (av)l
adjoint matrix of D( f ). can be evaluated by a O(log, n + a), as it can the operator Nf as

(4) The entries au of the matrix A(f) are polynomials of the ring R[Xl, . . . ,X,] having degree at most (n- l)(d - 1). Moreover, the Jacobian determinant J(f) is a polynomial

hf. Giusti et al. I Journal of Pure and Applied Algebra 1178~ 118 (1997) 277-317

303

of R[Xl, . . . ,X,] having degree at most n(d - 1). For 1 5 i 5 n we consider

Si

:=J(fWi

-

2

ai,jh.

j=l

All polynomials

appearing

have degree bounded

on the right-hand

side of the definition

of gi as summands

by v := nd + 1. Thus the degree of any gi is bounded

by v. Let

, . . . ,X,) E R[&, . . . ,X,] be the homogenization of gi by a new variable X0 . . . , X,) E R[Xo,. . .,X,1 be the homogenization of the Jacobian and let ‘J( f&Y&, determinant J(f) by X0.

h.4i(X0,XI

We introduce now the following homogeneous polynomials (forms): - Gi(Xo,. ,A’,) :=X~-deg(gz)(hgi), ~ N(&, . . . ,X,) := X;-deg(J(f))(hJ( f)). According to Lemma 27, there exists a division-free straight-line program

in R[X,,

. . .,X,,] of size 0(d2(n7 + n3L)) and non-scalar depth O(log, n + e) which evaluates the forms G1 , . . , G,, H. We now define recursively the following polynomials: - for k=l, l 2, 1 5 i < n let gIk’ := Gi(hck-‘), g(lk-‘I,. . . , &,k-l’), hck) := H(h(k-‘),g~-‘),

. ..) gn(k-1) ). It is easy to see that these polynomials nomial

hCk) is a denominator

gik’, . . . , gy’ are numerators and that the polyof the iterated Newton-Hensel operator Nj. A straight-

line program evaluating them is obtained by iterating k times the straight-line program which computes G,, . . . , G, and H. No new parameters are introduced by this procedure. Putting all this together we obtain the complexity bounds in the statement of Lemma 30. 0 3.2. Lifting jibers by the symbolic Newton-Hensel The idea of using a symbolic was introduced algebraic lifting

in [25]. For technical

parameters

algorithm

adaptation

for the lifting

algorithm

of Newton-Hensel

reasons, process.

for lifting fibers

in this paper it was necessary

We present

in which the use of algebraic

iteration

numbers

trix with integer entries. The whole procedure therefore The new lifting process is described in the statement

here a new version is replaced

to use of this

by a certain ma-

becomes completely rational. of the next theorem and its

proof. Let notations and assumptions be as the same as at the beginning of this section. We fix 1 < i 5 n and assume for the sake of notational simplicity that the variables Xl,. . . , X, are already in Noether position with respect to the variety K, the variables Xl,. .,X,_, being free. We suppose that the lifting point fi, the coordinates of the Z-linear form Ui and a geometric solution for the equations of the lifting fiber Ve are explicitly given. With these conventions we state the main result of this subsection as follows.

M. Giusti et al. I Journal of‘ Pure and Applied Algebra 117& 118 (1997) 277-317

304

Theorem 31. There exists a (division-free) straight-line program & in the polynomial

ring Z[X, , . . .,X,_,, Uj] of size (id&L)‘(‘) and non-scalar depth O((log, i + k) log, Si) using as parameters - the coordinates of S, _ the integers appearing in the geometric solution of the equations of the lofting fiber VI, and - the parameters of the input program T such that the straight-line program C computes _ the minimal polynomial qi E Z[X,, . . . , Xn-i, Ui] of the primitive element ui of the ring extension Q[Xr , . . . , ~~-il~~[~l~~~xl~~~~~~~ll~fl~~~~~~~~ _ polynomials p”:. , . . . ,Xn_i], p := nL=,_,, pf’ and polynomials n ,+,,...J)EZ[Xl n v(l) ,,+,, . . , vf’ E Z[Xl,. . ,Xn-i, U,] with max{degurvj’); r
-V~~I(Vi),...,p~‘X,-V~‘(Uj))p

holds. Without loss of generality, we may assume that c represents the coeficients of the polynomials qi and vf$, , vt’ with respect to Ui. Proof. Under our hypotheses, namely that XI,. . . , X, are in Noether position with respect to the variety K, the variables Xl,. . . ,Xn_i being free, we have the following integral ring extension

of reduced rings:

Ri I= O[X, ,...,X,-i]--tBi:=~[X,,...,X,]/(fi,...,J;). Let fl=( ~1,. . . , pn-i)E Znpi be a lifting point of the morphism rc, and let V, = ni’(i9) be its lifting fiber. We have deg V, = Di = rankRZ Bi 5 deg J$ = 6i. Let Vi = &_i+rXn-i+t + ” ’ + 2,X, E Z[Xn-ifI7 . . .,X,1 generate a primitive element of the ring extension Q +Q[F$] (i.e., Ui separates the points of V,). The minimal polynomial of the image of Ui in Bi has degree at least the cardinality of the set Ui(Vc). Since the linear form Ui separates the points of I$, this cardinality is deg Ve = rankR,Bi. We conclude that

U, generates also a primitive

element of the ring extension Ri+Bi. In the sequel we denote the primitive element generated by the linear form Vi in the ring extensions Q-tQ[V~] and Ri -fBi by the same letter ui. By hypothesis

the following

data is given explicitly

(i.e., by the bit representation

of their coefficients): - the primitive minimal equation q E Z[T] of the primitive element u; of Q[b], _ the parametrization of Vc by the zeroes of q, given by the equations Xr - p1 =O,...,Xn__i

- Pn__i=O,

4(T) = 0,

-v~-i+l(T)=O,...,p,X~-v~(T)=o,

Pn-i+lXn-i+l

. . , v,, are polynomials of Z[T] of degree strictly less than Di = deg q integers. Check again Definition 10 to see that the polynomials pn-i+lXa-i+t - an-i+r( T), . . . , p,X,, - vn(T) are assumed to be primitive. Let a E Z be the leading coefficient of q. where

t+-i+l,

.

and on-i+l,..., pn are non-zero

305

M Giusti et al. I Journal of Pure and Applied Algebra I17& 118 (1997) 277-317

We consider elements

now

fi,. ..,fias polynomials

of the polynomial

be the corresponding

Recall that the Newton

. . ,X,1. Let

ring Ri[X,_i+l,.

Jacobian matrix off operator

in the variables

Xn_i+l,. . .,X,,, i.e., as

f :=(fi , ...,f;)and let

(with respect to the variables XnPr+r,. . . ,X,).

with respect to these variables

is defined as

Using Lemma 30, we deduce the existence of numerators ~~_~+l,. . . , g,, and a non-zero denominator h in the polynomial ring Ri[Xn_i+l,. . . ,X,] such that the following K-fold iterated Newton operator has the form

Let K be a natural number.

cl.-,+1

h

Iv;=

0 :

.

9_ h

Now,

let MC_,+,

Q-algebra

, . . . ,Mx, be the matrices describing the multiplication tensor of the Q[Q] (recall that by assumption a geometric solution of the polynomial

equation system defining are known).

Vj is given and that therefore

the matrices a-‘q(

Let M denote the companion matrix of the polynomial 1
Moreover,

T) E Q[T}.

Let n - i +

Vj(M).

the matrices

pj#l-‘Mx,

have integer entries. Let K := [l + log, Sil and note

that K 2 1 + log, Di holds. Our straight-line a subroutine

I&_,+, , . . . , Mx,

program

fi will execute K Newton steps in

which we are going to explain now: Let us consider the following

column

vector of matrices Nn_i+r,, . . , N,, with entries in Q(Xr,. . . yXn_i):

(?+j

:=&-f(M)=

(‘“;z’), > n

whereM = Wfx,_,+, , . . . ,Mx,) and gn-i+l, nator polynomial

of Lemma 30. Finally,

A := Ui(Nn-i+r,.

I,_

. . . , g,, are the numerator

let us consider

. . , N,) = jbn_i+lNn-i+l

+

and h the denomi-

the matrix

+ A,Nn

This matrix is a matrix whose entries are rational functions of C&Y,, . . .,X,-i). From the fact that P,=(p,,..., pn_i) is a lifting point and from the proof of Lemma 30 one deduces easily that in fact the entries of A’Z belong to the local ring Re :=(Ri)(~,--pl,...,

~,_,-p,_,)=a[x,,...,x,-il(x~-p,

,,.,,x,-,-p._,).

M. Giusti et al. I Journal of Pure and Applied Algebra 1176~ 118 (1997) 277-317

306

Let T be a new variable.

With these notations

and assumptions

we have the following

result:

Lemma 32. Let xEQ(X,,..., Xn_i)[T] be the characteristic polynomial of A%?and m, E Ri[T] the minimal integral equation of the primitive element ui of Bi over Ri. Let x(T) = TDl+CfAi’ akTk and m, = Tol+C~~~’ bkTk with ak, bk E 0(x,, . . ,Xn-i) for 0 di+l, where orde denotes the usual order function (additive valuation) of the local ring R8. Proof. Let V, = {
fiber V, and from Hensel’s

Lemma

(which

represents

a

symbolic version of the Implicit Function Theorem) we deduce that there exist formal power series Rf!,,, , . . . , R~~‘EC[[X~ - pl,...,X,_i - pn-i]] with R~~i+l(P)=~~)i+l,

. . ..R.)(P;)=@

such that for R(‘):= (XI - p,, . . .,X,_,

- pn_-i,RyLi+,, . . . , RLt’) the

identities

f,(R’t’)=O ,.._,. fi(R(t))=O

(5)

hold in @[[Xl - PI , . . . ,X,_, - pnJ]. Let z4(‘):= L’i(R(‘)) = An-i+lRfl,‘li+l+ . . +A,R$“. As shown in 1251, the minimal polynomial m, of the primitive element ui verifies in @[[Xl - ~1,. . ,Xn_i - p,_i]][T] the identity

m, = II

(T - u(t)).

(6)

above which produces the matrix J@ E (Re )D1xD3 starting with the matrix M E QD3xDC (recall that M is the companion matrix of the polynomial a-‘q(T) E Q[T]). The same construction transforms the Di distinct eigenLet us now consider the construction

values of the diagonalizable matrix M, namely the values c, = Ui(tl), 1 < 15 Di, into eigenvalues of JY. (Observe that by construction 4 is a rational function of the matrix M and therefore the same rational function applied to any eigenvalue of M produces an eigenvalue of JY.) As shown in [25], in this way we obtain Di distinct rational functions zZ(‘)E @(Xl,. . . ,Xn-i), which are eigenvalues of ~4’. Moreover, these rational functions are all defined in the point Pi (this means that u”(‘)E @[Xl,. . .,X,,_&_, ,,,,,,x,,_,_~+,) holds). For 1 i: 1
- pn-;)6S+‘.

- p,,-i]]

and they satisfy in this

(7)

(For a proof of these congruence relations see [25].) Let us now consider the characteristic polynomial x of the matrix J%‘. Since the coefficients of JY belong to RF:= O[Xl , . . . ,xn-il(x,-p,,...,x,~,-p,_,), the c~~~cient~of x do so too. Therefore, x can be interpreted as a polynomial in the variable T with coefficients in the power series ring a[[& - ~1,. . ,Xn-i - pm-i]]. From the fact that the

M. Giusti et al. I Journal of Pure and Applied Algebra 1 I7& 118 (1997) 277-317

rational

u”(l), . . . , iPI)

functions

x(T)=

J-J I
eigenvalues

of J&’ we deduce that (8)

the k-th elementary

symmetric

function

in Di arguments.

(8) implies that for 0 < k < Di - 1 we can write the coefficient

of the polynomial

relations

m, satisfies bk = (- 1 )“‘-kek(U(‘),

(7) and the identities

the congruence

. . . , z&@)). From the congruence

(6) and (8) we conclude

now that for 0 5 k 5 Dj - 1

relations

ak - bk E (xl - p1,. . . ,Xn-i - Pn_i)6f+’ hold in O[[Xi - ~1,. . . ,Xn_i - p,_i]]. We continue coefficients

The

ak of x( 7’) as

ii@~)). From the identity (6) we deduce that the kth coefficient

ak=(-l)D’-kok(U”“‘,..., bk

D, distinct

(T-G(“)

holds. Let csk denote identity

represent

307

This proves the lemma.

now the proof of Theorem

of 1, let us consider

(9) 0

3 1. In order to see how to evaluate

6 := det (h(g))

the

E Ri and the matrix

A!, := 9./Z. This matrix -Ai has entries in Ri. Now we have the identities det(T-IdD, Let &T)=TDf

- A)=det(T.Id~~

- 8-1~,)=8-D’det((BT)Id~,

-&I).

+ $D,-~T~~-’

-A$. From the identities of x can be written as

+ .. . + ~$0E Ri[T] be the characteristic polynomial of above we deduce that for 0 5 k 5 Di - 1 the kth coefficient

Executing K = [l + log, Sil steps in the Newton iteration (as given at the beginning of this proof) to produce the entries of the matrix M, 8, JZ and finally Ai (in this order) and applying non-scalar parameters

Lemma 24 we produce

a straight-line

program

q! in Q(Xi,. . . ,Xn-i)

of

size O(log, &d3i6L) and non-scalar depth O((log, i + /) log, Si) using as those given by the statement of Theorem 31 such that 4’ evaluates the

family of polynomials 1,8,. . . , BDs, 40,. . ,40, -1 E Ri. Now, applying Strassen’s Vermeidung von Divisionen technique (Proposition 23) we obtain a division-free straight-line program 4.” in Q[A’l, . . . ,Xn-i] of size (idGiL)‘( non-scalar depth O((log, i + t) log, Si) using as parameters the coordinates pi,. . . ,pn-, of P,, the coefficients of the linear change of coordinates for the Noether normalization for E, the rational numbers appearing as coefficients in the geometric solution of the lifting fiber Vp8and the parameters of r such that 4” evaluates for each 0 5 k < D; - 1 the expansion of ak in Q[[Xi - ~1,. . .,X,_, - p,_,]] up to terms of degree of order 6; + 1. Taking into account the congruence relations (9) we see that the division-free straight-line program ry evaluates polynomials go, . . , 90, _ 1 E fil![&, . . . ,Xn-i] such that bk - gk E (Xl -

pl,.

. . ,Xn_j - Pn-i)“+’

308

M.

Giusti et al. IJournal

holds in O[[Xi - pi,. . ., X,-i that the degrees

of Pure and Applied Algebra 117&118 (1997) 277-317 - Pn-i]]

of the coefficients

exceed 6;. Putting

all this together,

for any 0 5 k < Di - 1. From [25] we deduce

bk of the minimal we conclude

polynomial

m, E Ri[r]

do not

that

bk=gk

holds for any 0 5 k < D, - 1. This means that the division-free

straight-line

program

4” computes the coefficients of the polynomial q1 := m, E Q[Xl, . . . ,X,, V;] = Ri[Ui]. In order to compute the parametrizations Ugly+,, , v!’ E O[X,, . . . ,X,_,, T] we have to use the same kind of techniques bined with the arguments

developed

(namely

truncated

Newton-Hensel

in [34] and applied

in [25,26].

iteration)

com-

Let us be more

exact. Let n-if1 5 k 5 n, let & be a generic linear combination of the variables & and Ui (one may use a new indeterminate for that) and iet rnx, E R,[Xk] and ??Q E Ri[Zk] be the minimal

polynomials

and zk in the Ri-algebra

of the Rj-linear

endomorphisms given by the images of & B, = Q[ K] = o[Xi, . . . , X,1/( f,, . . , fi ). The polynomials mx,

and mz, are manic and hence squarefree. We compute the coefficients of the polynomials mx, E Ri[Xk] and mz, E Ri[Zk] in the same way as before the coefficients of qi = m,. So, we have two manic squarefree polynomials mxk E Rj[Xk] and rnzk E R;[Zk]. Taking into account that by the choice of zk the variable Ui is a generic linear combination of & and zk and therefore separates the associated primes of the ideal (mx,, mz, ) in R;[Xk,&] = Ri[Xk, Ui] we can apply directly Lemma 26 in [34] in order to obtain the parametrization

associated to the variable xk. Doing the same for each of the variables involved, namely the variables Xn_i+, , . . . , X,, and putting together the corresponding straight-line programs, we obtain a procedure and a straight-line program & of the desired complexity which computes the output of Theorem 3 1. 0

3.3.

The recursion

Proposition 33. There exists a division-free (id6iL)O(‘) and non-scalar depth O((log, i-t/)

arithmetic network of non-scalar size log, Si) using parameters of logarithmic

height bounded by max{h, r/i, O((log, i + e) log, Si)} which takes as input _ a Noether normalization for the variety V:, _ a lifting point Pi, and _ a geometric solution of the lifting ,fiber VP, and produces as output _ a linear change of variables (Xl,. .,X,,) + (Yl(‘+“, . , Y,(‘+‘)) such that the new variables Y[‘+‘), . . . , Y,(‘+‘) are in Noether position with respect to E+l, _ a lifting point Pi+, for E+I, and _ a geometric solution for the lifting ftber VP,,,. Proof. The construction of the arithmetic network proceeds in three stages. In the first stage we apply the algorithm underlying Theorem 31. In the second stage we intersect algorithmically the variety K with the hypersurface V(f;+,) in order to produce first a

M. Giusti et al. IJournal of Pure and Applied Algebra 117&118

Noether normalization Then

we produce

of the variables

a linear

with respect to the variety

U;+l representing

form

Ri+l -+ Bi+l and a straight-line

integral ring extension analogous

to the one in the conclusion

algorithm

underlying

of Theorem

the proof of Proposition

take care of is that we are working

a primitive

277-317

309

K+l = E n V(fi+l). element

ui+l

program representing

of the

polynomials

31. For this purpose

we use the

14 in [25]. The only point we have to

now over the ground field Q and that we have to

take into account the heights of the parameters program. We remark that the straight-line

(1997)

program

of Q we introduce

in this straight-line

in question

has non-scalar

size (id6J)0(‘)

+ 6) with parameters

of logarithmic

height bounded

and non-scalar

depth O(log,(dGi)

by O(log2(dGi)

+ e). In the third and final stage we consider

the polynomial

JU’I . . ..h+~> := det the ideal Ii+1 = (fi,. . . , fi+, ). Let us observe that

which is a non-zero

divisor modulo

the polynomial

. . . , fi+l ) can be evaluated

J(fi

by a division-free

straight-line

of length O((i + 1)5 + L) and depth O(log,(i + 1) + S). Let ~1E a[& , . . . ,Xn-i_ I] be the constant term of the characteristic

program

polynomial

of

the homothety given by J := J(f, , . , fi+l ) modulo Ii+,. Since J represents a non-zero divisor modulo Zi+, we conclude that /A does not vanish. Furthermore, we observe that 1-1can be evaluated by a division-free straight-line program in O[Xl,. . . ,Xn_i_1] of length (i6iL)O(‘) and depth O(log, d6i + (log, i + t) log, Si) = O((log, i + t) log, S,), and so does the product p . p, where p = nE_,+, pi” is defined as in the statement of Theorem 3 1. Using

a correct test sequence (see [32]) we are able to find in sequential time and parallel time O((log, i + d) log, Si) a rational point Pi+1 E .Zn-i-l of logarithmic height bounded by O((log, i + C) log, Si) which satisfies (p.p)(Pi+l) # 0.

(i6L)“(”

Clearly,

P,+l

is a lifting

point

for the variety

F+l.

In order to obtain

a geometric

solution of VP,+,with primitive element ui+l induced by the linear form Ui+l we have to specialize in the point Pi+, the polynomials obtained as output of the second stage (see [25, Section 31). By this specialization

we obtain the binary

representation

of the co-

efficients of certain univariate polynomials in Ui+l which represent a geometric solution of the fiber VP,,, . Nevertheless, it might happen that the height of these coefficients is excessive. In order to control the height of these polynomials we make them primitive. This requires some integer greatest common divisor (gcd) computations which do not modify the asymptotic

time complexity

of our algorithm.

0

4. Lifting residues and division modulo a complete intersection This section is dedicated to the proof of Theorem trace formula for Gorenstein algebras given by complete

ideal

21. The outcome is a new intersection ideals. This trace

M. Giusti ec al. I Journal of Pure and Applied Algebra 117& 118 (1997) 277-317

310

formula does not make reference Our trace formula represents

anymore

to a given monomial

an expression

basis of the algebra.

which is “easy-to-evaluate”.

4.1. Truce and duality Trace formulas mination

theory.

appear in several recent papers treating problems Some of these papers

use a trace formula

in algorithmic

eli-

in order to compute

a

quotient appearing as the result of a division of a given polynomial modulo a given complete intersection ideal (see [22,34]). Other papers use trace formulas in order to design algorithms for geometric (or algebraic) solving of zero-dimensional Gorenstein algebras

given by complete

intersection

ideals [ 1,2, 141. The paper [49] uses a trace

formula to obtain an upper bound for the degrees in the Nullstellensatz. However, all these applications of trace formulas require the use of some generating family of monomials

of bounded degree which generate the given Gorenstein

algebra as

a vector space over a suitable field. As a consequence, such trace formulas provide just syntactical complexity or degree bounds and in particular no intrinsic upper complexity bound (as e.g., in Theorem 21) can be obtained in this way. In this subsection we introduce an alternative trace formula in order to obtain maximum benefit from the geometrically and algebraically well-suited features of Gorenstein algebras. Let us start with a sketch of the trace theory. For proofs we refer to [37, Appendices E and F]. Let R be a ring of polynomials over a given ground field (for our discussion the ground field may be assumed to be Q). Let K be the quotient field of R and let . , fn R[& , . . . ,X,] be the ring of n-variate polynomials with coefficients in R. Let fi, be a smooth regular sequence of polynomials in the ring R[Xl, . . . ,X,,] of degree at most d in the variables Consider

now the R-algebra

B := R[X, We assume

Xl, . . . , X,, generating

a radical ideal denoted by (f;,

B given as the quotient

of R[X,, . . . ,X,] by this ideal:

,...,x,ll(fl,...,h>.

that the morphism

R + B is an integral

ring extension

representing

Noether normalization of the variety V( fi , . . , fn) defined by the polynomials in a suitable affine space. Thus, B is a free R-module of rank bounded by the the variety V( fi,. ..,fn)(this estimation is very coarse but sufficient for our Moreover, the R-algebra B is Gorenstein and the following statements are this fact. We consider B* := HomR(B,R) BxB*

..,fn).

as a B-module

a

fi , . . . , fn degree of purpose). based on

by the scalar product

-+B*

which associates to any (b,z) in B x B* the R-linear map b.z : B +R defined by (b.z)(x) := z(bx) for any element x of B. Since the R-algebra B is Gorenstein, its dual B* is a free B-module of rank one. Any element a of B* which generates B* as B-module is called a trace of B. There are two relevant elements of B* that we denote by Tr and a. The first one, Tr, is

M. Giusti et al. IJournal of Pure and Applied Algebra 117& 118 (1997)

called the standard trace of B and it is defined in the following qb : B + B the R-linear image Tr(b)

map defined by multiplying

311

way: given b E B, let

by b any given element

under the map Tr is defined as the ordinary

qb of B (note that this definition

277-317

of B. The

trace of the endomorphism

makes sense since B is a free R-module).

In order to

introduce

c (which will be a trace of B in the above sense), we need some additional

notations.

For any element

g E R[Xl , . . .,X,1

we denote by S its image in B, i.e., the

residue class of g modulo the ideal (f,, . . . , fn). Let Y,, . . . ,Y, be new variables and let Y :=(Yl , . . .,Y,). Let 1 < j 2 n and let 4’ := A( Y,, ,Y,) be the polynomial of R[YI,..., Y,] obtained by substituting us consider the polynomial

in 4 the variables

.f,‘-L=~G(Yk-Xk)ER[~l>...>

&,Yl>...>

Xi,. . . , X, by Y,, . . . , K. Let

r,l,

k=l where the bk are polynomials at most (d - 1) (observe

belonging

that the

J!jk

to R[&, . . . , X,,, Yi, . . . , Y,] having total degree are not uniquely

J;,..., fn). Let us now consider the determinant can be written (non-uniquely) as A=

determined

A of the matrix

by the sequence (Ijk)isj,kln

which

Ca,l(x,,...,X,)b,(Y,,...,Y,)EW[~~,...,X,,Y,,...,Y,1, m

of R[Xl , . . . ,X,] and the b, elements of R[ Yl, . . . , K]. (Observe that it will not be necessary to find the polynomials a, and b, algebraically, we need just their existence for our argumentation.) The polynomial A is called a pseudo-

with the a,,, being elements

of the regular sequence (ft , . . . , fn). Observe that the polynomials a, and b, can (and will) be chosen to have degrees bounded by n(d - 1) in the variJacobian determinant

ables Xi,. .,X,, and Yi,. . . , K, respectively. Let c, E R[Xl,. .,X,1 be the polynomial we obtain by substituting in b, the variables Yi,. . . , Y, by Xl,. . . ,X,. For j the class of the Jacobian

determinant

J(fi,

. ..,fn)in B we have the identity

the image of the polynomial A in the residue class ring R[X,,. . . ,X,, , K] modulo the ideal (f, , . , fn, fiy, ,f,' ) 1sindependent of the particular choice Of the matrix This justifies the name “pseudo-Jacobian” for the polynomial A. With these notations there exists a unique trace rs E B* such that the following identity holds in B: Moreover, Y1,

(t!kj)l
The main property of the trace 0, known as “trace formula” (“Tate’s trace formula” [37, Appendix F], [38] being a special case of it) is the following statement: for any gER[Xl,...,X,] the identity

(10)

M. Giusti et al. IJournal of‘ Pure and Applied Algebra 117&118

312

holds true in B. Let us observe that the polynomial underlying

the identity

Problem 34 (Lifting degree

a(g . a,) . c, E R[Xl, . . . ,X,]

C,

(10) is of degree at most n(d - 1) in the variables

The main use of this trace formula consists

bitrary

(1997) 277-317

of a residual

in Xi,. . .,X,,

class).

in solving

the following

Given a polynomial

find a polynomial

Xi,. . . ,X,.

problem.

g E R[Xi, . . . ,X,]

of ar-

g1 E R[Xi, . . . ,X,J of degree at most

Xi,. . .,X,, such that j, = 3 holds in B.

n(d - 1) in the variables

As we have seen before, the trace formula

( 10) solves Problem

34 since it allows

us to choose for gi the polynomial

c

g1 :=

[email protected],). c,

(11)

m

However, advantage

defining the polynomial g1 by the formula (11) inhibits us from taking of any special “semantical” features of the R-algebra B: one “a priori” needs

all monomials of degree at most n(d - 1) for the description of the polynomials c, (and a,). Therefore, we replace the trace formula (10) by the following alternative one. Proposition 35 (Trace formula).

With the same notations as before, let us consider

the free R[Xl, . . . ,X,,]-module B[Xl , . . . ,X,,] given by extending scalars of B (this means we consider the tensor product B[X,, . . ,X,,] := B @R R[Xl,. . . ,X,1) and let us also consider the polynomial Al E R[X,, . . . ,X,,] given by AI :=~&+,EB[X m

,,...,

Then for any g E R[Xl , . . . ,X,]

X,]. the following identity holds true in R[X,, . . . ,X,,]:

Ca(Y.a,).c,=T;(J-‘S.A,). m (Here Tr := Tr C%IdR[x ,,,._, x,,l : B[X,, . . . ,X,] + R[X, , . . . ,X,]

is the standard trace ob-

tained from the standard trace Tr : B + R by extending scalars). Proof. Let Tr : B -+ R be the standard trace of the free R-module [37, Appendix Tr(J-‘j)

B. Let us recall from

F] that for any g E R [Xl,. . ,X,J the identity = o(j)

holds. From the R[ Y,, . . . , Y,]-linearity of the map 6 : B[Yl,..., deduce that any g E R[Xl , . . . ,X,J satisfies the identities %(?‘gA1(Y,,...,

K)) = C Tr(J-‘Sa, m

In other words we have in R[Y,, . . . , K] Tlr(J-‘gA,(Y~,...,~~))=

x+j.&).b, m

.b,)=

K] + R[Yl,. .., K] we

xT;(j-‘j&&b,. m

313

M. Giusti et al. I Journal of Pure and Applied Algebra 117& 118 (1997) 277-317

for any g E R[Xl , . . . ,X,1. Replacing

in this identity the variables

Yt , . . . , K by Xl,. . . , A?,,

we obtain the desired formula Tr(J-‘&I,)=

~f7((j.am),c,. m

0

One easily sees that for any h E R[X,, . . . , X,,, Y,, . trace of the image h of h in the R[ Yl, .

, X,], T?(h) is simply the standard

, Y,]-module

B[ Y,, . . . , G]. This observation

together with Proposition 35 represents our basic tool for the evaluation of formula (11) and hence for the solution of Problem 34. This is the content of the following considerations. Let us consider

the K-algebra

B’ = K @R B obtained

by localizing

B in the non-zero

elements of R. Fix a basis of the finite dimensional K-vector space B’. Let M., , . , Mxn be the matrices of the homotheties ylx, : B’ +B’ with respect to the given basis of B’ and let Tr denote the function which associates these conventions let gt be defined as 91 :=Tr(J(fl,...,

fn)(&,...,&)-’

to a given matrix its usual trace. With

.g(Mx I,..., Mx,)

.d(Mx,,...,Mx,,Xl,...,X,)).

One easily verifies that gr belongs

(12)

to R[Xl,. . .,X,1 and that jr = S holds in B.

4.2. A division step The lifting process presented in the last subsection is now applied to compute the quotient of two polynomials modulo a reduced complete intersection ideal. More precisely, let us consider a polynomial f E R[Xl , , . .,X,,] which is not a zero-divisor in 3 and another polynomial g E R[X, ,. . . ,&I such that the residue class y divides the residue class j in B. The following proposition shows how we can compute q E R[X, , . . . ,X,J for the division of j by f in B.

a lifting

quotient

Proposition 36 (Division step). Let notations and assumptions be the same as in the previous subsection and let D be the rank of B as free R-module, Let be given the following items as input: _ a straight-line program S of size L and depth e representing the polynomials .f,g and fi,...,fn; _ the matrices Mx, , . . . , Mx, describing the multiplication tensor of B with respect to the given basis of B’ = K @R B. Suppose that f is a non-zero divisor of B and that f divides g in B. Then there exists a division-free straight-line program r in K[Xl,. . . J,,] of size L(ndD)‘(‘) and non-scalar depth O(&log, D+log, n j which computes from the entries of the matrices Mx, , . . . , Mx, and the parameters of r’ a non-zero element 0 of R, and a polynomial q of R [XI,. . . ,X,,] such that 0 divides q in R [XI , , . . ,X,,] and such that qf = &j holds in B.

314

hf. Giusti et al. I Journal of Pure and Applied Algebra 117& 118 (1997) 277-317

Proof. In order to prove this result, let us observe that any basis of B as free R-module induces

a basis of B[Yl, . . . , &] as free R[Y,, . . . , Y,]-module.

matrix of the multiplication as well the multiplication

if Mx, is the

by Xi in B[Yl, . . . , X] with respect to the same basis. Next, and J(fi, . ..,fn)are not zero-divisors modulo (fi, . . . , fn),

since the polynomials

f

the following

are non-singular:

matrices

Moreover,

by Xi in B with respect to a given basis, Mx, represents

FI :=fW..,,...,Mc,),

Finally,

let us denote by Gi and Al the following

two matrices:

and

AI :=A(Mx,,...,Mx,Y,,...,Y,), where A is the pseudo-Jacobian

determinant

of fi, . . . , fn. Let us remark that the matri-

ces F,, J1 and Gi have entries in K while Al has entries in K[Yi, . . . , Y,J. From formula (12) of the previous subsection we deduce that q1 :=Tr(Jy’ . F,’ . Gl . Al(X,, . . . , X,,)) S (by Tr we is a polynomial of R [Xl,. . . , X,] which satisfies in B the identity qij= denote here the usual trace of matrices). Finally, let us transpose the adjoint matrices Fz := ‘Adj(Fl), The quotient

and

of F1 and J1:

Jz := ‘Adj(Ji).

q E R[X,, , . . , X,,] and the non-zero

constant

0 E R we are looking

for are

given as follows: - q:=Tr(F2.J2.G1.Al(Xl,...,Xn)), - 19:= det(F2).

det(J2).

Clearly, the polynomial

q1 is the quotient

of q/8, while gj=

over, q can be computed

by a division-free

straight-line

&j holds in B. More-

program

r

in K[&, . . . ,X,,]

from the entries of Mx,, . . . , Mx, and the parameters of r’ (note that for the computation of q1 we need divisions). The complexity bounds in the statement of Proposition 36 follow by reconstruction of the programs determinants involved (cf. Section 3.1).

evaluating

f, g, J(f,,

. . . , fn), A and the

Proof of Theorem 21. In order to prove Theorem 21 we follow the algorithm underlying the proof of Proposition 36. We just have to add just some comments conceming the matrices Mx,, . . , Mx,. Let us consider a linear form ,4,X, + . . +1,X, (with %iE Z) inducing

a primitive

element u of the zero-dimensional

Q-algebra

Q[Xi , . . . ,X,1/

(f,, ...,fn).In this case R will be the field Q. Let qu E Z[T] be the minimal polynomial of u and pi&

- vi(T), . . . , p,,X, - v,(T)

the parametrizations

of the variety with

M. Giusti et al. IJournal of Pure and Applied Algebra 117&118

respect to this primitive a discriminant.

element.

Let a be the leading coefficient of q,, and p = n:=,

where A4 is a matrix with integer

entries. Now, the matrices

cation tensor of U&Y,, . . .,X,,]/(f,,.

. . ,fn)

,X,]

the multipli-

can be written as

andf=f,+r

in Proposition

36 we obtain 0 E Q, 8 # 0 and

such that

Q. 1 -q..fn+1

E(fl,...,hz)

holds. Finally, multiplying integer a and a polynomial a:=

describing

Zli(X-‘M)

for 1
pI

matrix of qu has the form

Then the companion

Mx, = pi’

315

(1997) 277-317

cxNpM.OEZ,

by appropriate powers g,+l of the form gn+l

such that a - g,+r fn+r E (ft,.

:=

of CI and p we obtain

a non-zero

8p”qEz[x,,...,Xn]!

, fn) holds. The bounds

of the Theorem

21 are then

obtained from the bounds of Proposition 36. The bounds for the height of the parameters are obtained choosing an appropriate primitive element u such that qu and the parametrizations

have height equal to the minimal

V := V(,fi ,...,fn>.

height of the diophantine

variety

0

Acknowledgements L.M. Pardo wants to thank the l?cole Polytechnique

for the invitation

and hospitality

during his stay in the Fall of 1995, when this paper was conceived.

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