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The Euclidean distance degree of Fermat hypersurfaces
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Journal of Symbolic Computation www.elsevier.com/locate/jsc
The Euclidean distance degree of Fermat hypersurfaces Hwangrae Lee 1 Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea
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Article history: Received 3 July 2015 Accepted 3 July 2016 Available online xxxx Keywords: Euclidean distance degree Fermat hypersurface Optimization
a b s t r a c t Finding the point in an algebraic variety that is closest to a given point is an optimization problem with many applications. We study the case when the variety is a Fermat hypersurface. Our formula for its Euclidean distance degree is a piecewise polynomial whose pieces are defined by subtle congruence conditions. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Let X ⊂ Rn be a real affine algebraic variety, i.e., X is the common zero set of some polynomials f 1 , . . . , f m ∈ R[x1 , . . . , xn ]. We consider the following problem: given u ∈ Rn , compute u ∗ ∈ X that n minimizes the squared Euclidean distance du (x) = i =1 (u i − xi )2 from the given point u. This problem arises from best approximation problems. Once we observe a data u in a mathematical model X by experiment, the data may be perturbed by some noise so that u does not actually lie on X . In such a situation, the nearest point u ∗ to u in X represents the original data suggested by u. In order to find u ∗ algebraically, we consider the zeroes in Cn of the equations defining X , and n we examine all complex critical points of the squared distance function du (x) = i =1 (u i − xi )2 on X \ X sing where X sing is the singular locus of X . If X has a positive dimensional singular locus, then there could be infinitely many critical points of du (x) on X . Thus we remove the singular locus of X . The number of such critical points is finite and constant on a dense open subset of data u ∈ Rn . That number of critical points was studied by Draisma et al. (2016). It is called the Euclidean distance
1
E-mail address: [email protected]. URL: http://myhome.postech.ac.kr/meso/. The author was supported by IBS-R003-D1.
http://dx.doi.org/10.1016/j.jsc.2016.07.006 0747-7171/© 2016 Elsevier Ltd. All rights reserved.
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degree (ED-degree) of the variety X , and is denoted as EDdeg( X ). From now on, all the objects will be considered as complex varieties, except in Section 2.3. Sometimes, X is given by homogeneous polynomials. The set of m1 × m2 matrices of rank at most k is a typical example. Such a variety is called a projective algebraic variety in Pn (C). For the definition of Pn (C) and more informations, see Chapter 8 of the book by Cox et al. (1992). For a projective X ⊂ Pn (C), we define EDdeg( X ) to be the ED-degree of the affine cone of X in Cn+1 . That is, just regard X as an affine variety and compute the ED-degree. The ED-degrees of determinantal varieties as above have been studied by Ottaviani et al. (2014). This paper is motivated by following general upper bound on the ED-degree. Proposition 1 (Draisma et al., 2016). Let X be a hypersurface in Pn (C) defined by a homogeneous polynomial f of degree d. Then
EDdeg( X ) ≤ d
n −1 (d − 1)i , i =0
and equality holds when f is generic.
2
In this paper, we focus on Fermat hypersufaces and their variations. Definition 2.
• A Fermat hypersurface of degree d in Pn (C), denoted by F n,d , is the projective variety defined by the polynomial xd0 + · · · + xnd . • An affine Fermat hypersurface of degree d in Cn , denoted by A F n,d , is the affine variety defined by the polynomial xd1 + · · · + xnd − 1. • A scaled Fermat hypersurface of degree d in Pn (C) with scaling vector a = (a0 , . . . , an ) ∈ (C∗ )n+1 , denoted by S F na,d , is the projective variety defined by the polynomial xd0 /a0 + · · · + xnd /an . 2 In statistical optimization, maximum likelihood estimation (MLE) is an important tool. The generic number of the critical points of the maximum likelihood function, called ML-degree, is a parallel concept to ED-degree. The ML-degrees of many statistically relevant varieties have been computed (Huh and Sturmfels, 2014). Recently, in particular, the ML-degree of F n,d is partially given by Agostini et al. (2015). Their results, which we review in Example 8, serve as motivation to our study of the ED-degree of F n,d . This paper is organized as follows. In Section 2, we will investigate the sharpness of the general bound (Proposition 1) for the Fermat hypersurfaces. We give a formula for the ED-degree of F n,d (Theorem 4), that gives an explicit formula for n ≤ 3 (Remark 5, Example 8). If we fix n and consider the general bound as a function in d, it is the best possible polynomial bound (Lemma 6), while the gap can be arbitrarily large (Remark 5). The main theorem can be used for an efficient algorithm which computes the ED-degree of a Fermat hypersurface numerically (Example 13). The proof of Theorem 4 can be used similarly to evaluate the ED-degree for an affine Fermat hypersurface A F n,d (Corollary 14). After that, an open problem (Conjecture 16) about real Fermat hypersurfaces will be discussed. In Section 3, we will consider the scaled Fermat hypersurfaces for fixed n and d. We introduce the exponential cyclotomic polynomial Q m, p which has a special role for the scaling vector a of S F na,d (Theorem 17). As a corollary, we will see that the ED-degree of a scaled Fermat hypersurface usually achieves the general bound of Proposition 1. 2. ED-degree for Fermat hypersurfaces 2.1. Main theorem for Fermat hypersurfaces In this section, we compute the ED-degree of F n,d for each n, d.
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Definition 3. For a positive integer p, fix a p-th primitive root of unity ζ . Define δ(m, p ) to be the number of integer m-tuples (t 1 , . . . , tm ), 1 ≤ t i ≤ p, satisfying
1+
m
ζ 2t i = 0,
i =1
or equivalently, satisfying that the univariant polynomial 1 + x2t i is a multiple of the p-th cyclotomic polynomial p (x). Note that it does not depend on the choice of ζ . 2 Theorem 4. For d ≥ 3, the ED-degree of the Fermat hypersurface F n,d is given by
EDdeg( F n,d ) = d
n −1 n n+1 · δ(m, d − 2). (d − 1)i − m+1 i =0
2
m =1
Theorem 4 will be proved further below. Remark 5. For small m, the following are derived easily from the definition:
(i)
(ii)
(iii)
δ(1, p ) =
2 if p ≡ 0 mod 4 0 otherwise
⎧ ⎨ 8 if p ≡ 0 mod 6 δ(2, p ) = 2 if p ≡ 3 mod 6 ⎩ 0 otherwise 12p − 24 if p ≡ 0 mod 4 δ(3, p ) = 0
otherwise
In particular, (iii) implies that the difference between the general bound and the actual ED-degree can be arbitrarily large. Although, the following lemma shows that the general bound is the best possible polynomial bound. 2 Lemma 6. If p is a prime bigger than m + 1, then δ(m, p ) = 0.
2
m
Proof. Assume 1 + i =1 ζ 2t i = 0. Replacing 2t i by 2t i − p if 2t i ≥ p for each i, we have a polynomial in ζ whose degree is less than p. Since p is a prime, it should be a scalar multiple of the cyclotomic polynomial p (ζ ) := 1 + ζ + · · · + ζ p −1 . It has p terms, hence m + 1 ≥ p. 2 No closed formula for δ(m, p ) is known, but it has been studied in both algebraic geometry and number theory (Adams and Sarnak, 1994; Laurent, 1984; Mann, 1965). In particular, a theorem given by Laurent (1984, Theorem 2) implies that δ(m, p ) is a polynomial periodic function in p in the sense of Adams and Sarnak (1994). Corollary 7. For fixed n, the ED-degree of F n,d is a polynomial periodic function in d. That is,
EDdeg( F n,d ) = for some periodic c i ’s.
c i (d)di
2
Example 8. Agostini et al. (2015) showed that the ML-degrees of the Fermat curves (n = 2) are given by
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⎧ 2 ⎪ ⎪ d2 + d ⎨ d +d−3 MLdeg( F 2,d ) = ⎪ d2 + d − 2 ⎪ ⎩ 2 d +d−5
if d ≡ 0, 2 mod 6 if d ≡ 3, 5 mod 6 if d ≡ 4 mod 6 if d ≡ 1 mod 6.
By the Theorem 4, we have
⎧ 2 d ⎪ ⎪ ⎪ 2 ⎪ ⎨d −2 EDdeg( F 2,d ) = d2 − 6 ⎪ ⎪ ⎪ d2 − 8 ⎪ ⎩ 2 d − 14
if d ≡ 0, 1, 3, 4, 7, 9 mod 12 if d ≡ 5, 11 mod 12 if d ≡ 6, 10 mod 12 if d ≡ 8 mod 12 if d ≡ 2 mod 12.
It is a polynomial periodic function in d, and the general bound EDdeg ( F 2,d ) ≤ d2 is the best possible polynomial bound. Comparing with MLdeg, both are periodic while their periods are different. 2 The system for critical points of the distance function is given by
xd0 + · · · + xnd = 0
(1)
xdi −1 (x j − u j ) = xdj −1 (xi − u i ) for each i = j
where the vector u = (u 0 , . . . , un ) ∈ Cn is sufficiently generic. The ED-degree is the number of solutions of (1) except (0, . . . , 0), which is a (unique) singular point of the affine cone over the Fermat hypersurface F n,d . Introducing a new variable t, we modify the system (1) into the following homogeneous system in S [t ] = C[x0 , . . . , xn , t ]:
S=
xd0 + · · · + xnd = 0
(2)
xdi −1 (x j − u j t ) = xdj −1 (xi − u i t ) for each i = j .
Each solution of S of the form (c 0 : · · · : cn : 1) corresponds to the solution (c 0 , . . . , cn ) of (1). The system S has more solutions that we do not want to count. Let mult (0) be the multiplicity of (0 : · · · : 0 : 1) for the system S, and (n, d) be the number of solutions of the form (a0 : · · · : an : 0) counting multiplicities. Then the ED-degree of F n,d is given by
EDdeg( F n,d ) = deg(S) − mult(0) − (n, d)
(3)
where deg(S) is the degree of the projective scheme defined by the system S. Now, Theorem 4 is just a consequence of following lemmas. Lemma 9. The multiplicity of (0 : · · · : 0 : 1) for the system S, denoted by mult(0), is d(d − 1)n .
2
Proof. Let I be the ideal in S [t ] generated by equations in S and m = (x0 , . . . , xn ) be the ideal corresponding to the point (0 : · · · : 0 : 1). Then mult(0) is defined by the length of S [t ]m / I m as an S [t ]m -module. In the local ring S [t ]m , the factor (xi − u i t ) is a unit. (By the genericity of u i , we may assume u i = 0 for all i.) Writing μi = (x0 − u 0 t ) · (xi − u i t )−1 , the localized ideal I m is generated by
xd0 + · · · + xnd = 0
xd0−1 = xdi −1 μi for each i .
(4)
Here, the length of S [t ]m / I m is just the maximum number of monomials in S which are independent α α modulo I m . By direct counting, we see that mult(0) = |{x0 0 · · · xn n | 0 ≤ α0 ≤ d − 1, 0 ≤ α1 , . . . , αn ≤ n ¯ ¯ d − 2}| = d(d − 1) . Alternatively, it is same as dimC ( S / I ) where I is the ideal in S defined by (4) after changing each μi into arbitrary nonzero value in C. Therefore, by Bézout theorem, we get the same answer. 2
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To compute (n, d) in (3), we want to put t = 0 in the system S to get
xd0 + · · · + xnd = 0
(5)
xdi −1 x j = xdj −1 xi for each i = j
This could give the wrong answer if S and the hyperplane t = 0 meet non-transversally. The next lemma shows that it is not the case. Lemma 10. The system S and the hyperplane t = 0 meet transversally. Hence (n, d) = deg((5)).
2
Proof. It suffices to show that the system S and ∂S/∂ t have no common root where ∂S/∂ t is the ideal generated by the t-directional partial derivatives of the equations in S. Let c = (c 0 : · · · : cn : 0) be a nonzero solution of S. It has at least two nonzero entries by the equation cd0 + · · · + cnd = 0. Without loss of generality, we may assume c 0 = 1 and c 1 = 0, which implies cd1−2 = 1. Also,
∂ d −1 x0 (x1 − u 1 t ) − xd1−1 (x0 − u 0 t ) |x=c ∂t = −u 1 + cd1−1 u 0 = 0 only if |u 0 | = |u 1 |.
By the genericity of u i , the last equality does not happen.
2
Lemma 11. Let δ(m, p ) be the function defined in Definition 3. Then we have
(n, d) =
n n+1 m =1
m+1
· δ(m, d − 2).
2
Proof. By Lemma 10, (n, d) = deg((5)). Let (c 0 : · · · : cn ) be a solution of (5). Suppose that c i = 0 for all i. Then x0 = 1 in the system (5) implies that all c i ’s are some (d − 2)-nd roots of unity. Fix a (d − 2)-nd primitive root of unity ζ , and write c i = ζit . Then the system (5) has δ(n, d − 2) many solutions. If a solution has m + 1 many nonzero coordinates, the number of such solutions is (for the choices of nonzero coordinates) times δ(m, d − 2). 2
n+1 m+1
Lemma 12. The degree of the system S is given by
deg(S) = d
n
(d − 1)i .
2
i =0
Proof. Let I be the ideal generated by
xdi −1 (x j − u j t ) = xdj −1 (xi − u i t ) for each i = j
= 2 × 2 minors of
xd0−1 · · · xnd−1 x0 − u 0 t · · · xn − un t
It defines a curve in PnC+1 . By Lemma 10, deg( I ) = deg( I + (t )). If c = (c 0 : · · · : cn : 0) is a solution for I + (t ), write cm = 1 where m is the first nonzero entry of c. For each nonzero entry except cm , there are d − 2 choices to be a solution. Hence the total number of solutions is
n +1 n+1 i =1
i
(d − 2)i −1 =
n (d − 1)i . i =0
Therefore deg(S) = deg( F n,d ) · deg( I ) = d
n
i =0 (d
− 1)i by Bézout. 2
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Proof of Theorem 4. We have
EDdeg( F n,d ) = deg(S) − mult(0) − (n, d). Apply Lemma 12, 9, and 11 to each term in the right side.
2
Example 13. We showed that the ED-degrees of the Fermat hypersurfaces can be computed by δ(m, p ) or (m, p ) without using the random data u. The following Macaulay2 (Grayson and Stillman, 1992) code computes the ED-degree of F n,d efficiently.
n=2,d=5; R=QQ[x_0..x_n]; gbd=sum apply(n,i->d*(d-1)^i); -- the general bound F=sum apply(n+1,i->(gens R)_i^2); M=matrix{apply(n+1,i->((gens R)_i)^(d-1))}||matrix{gens R}; I=ideal(F)+minors(2,M); EDdeg=gbd-(degree I)*(dim I) The output reveals that the Fermat quintic cone F 2,5 has ED-degree 23.
2
2.2. Affine Fermat hypersurfaces Let X be an affine Fermat hypersurface A F n,d . The system for the critical points of the distance function is given by
xd1 + · · · + xnd = 1
xdi −1 (x j − u j ) = xdj −1 (xi − u i ) for each i = j and the homogenized system is
xd1 + · · · + xnd = t d
(6)
xdi −1 (x j − u j t ) = xdj −1 (xi − u i t ) for each i = j
In this case, (0 : · · · : 0 : 1) is not a solution for (6) (see Lemma 9). Except that, the ED-degree of A F n,d can be computed in the same way as in the homogeneous cases. Corollary 14. For d ≥ 3, the ED-degree of the affine Fermat hypersurface A F n,d is given by
EDdeg( A F n,d ) = d
n −1
(d − 1)i −
i =0
n −1 n · δ(m, d − 2). m+1
2
m =1
Note that the first summand is the general bound for affine varieties, given by Draisma et al. (2016, Corollary 2.5). 2.3. Real critical points For odd d, the Fermat hypersurface F n,d can be considered as a nonempty real variety. In this case, the number of the real critical points of the squared distance function highly depends on the location of the given point u ∈ Rn+1 . Nonetheless, the next theorem gives an upper bound for the maximum possible (finite) number of the real critical points. By possible number we mean that there is an open set in Rn+1 of data space such that the number is attained for all points in the open set.
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Theorem 15. For the Fermat hypersurface F n,d , the number of the real critical points of the squared distance function is bounded by
√
2
25n2 −3n+2
· (n + 2)5n
2
Proof. Let u ∈ Rn+1 be a point not in F n,d , whose entries are all nonzero. Then the critical equation (1) can be written by
xd0 + · · · + xnd = 0
xdi −1 (x0 − u 0 ) = xd0−1 (xi − u i ) for each i = 1, . . . , n. This system has n + 1 polynomials in n + 1 variables, and the number of monomials used in this system is 5n + 1, which does not depend on d. By Khovanskii’s fewnomial bound (Khovanskii, 1980), this system has at most
2
5n 2
· (n + 2)5n
positive solutions. It is also an upper bound for the number of real solutions in any orthant, and there is no solution with 0 in its coordinates. Hence we can have at most
2n+1 · 2
5n 2
· (n + 2)5n
real solutions in total.
2
Note that this bound does not depend on d, hence we can ask for the sharp bound for each n. For n = 1, the real cone of F 1,d is a straight line in R2 , hence the critical equation has one real solution. We numerically computed the numbers of real solutions for d = 3, 5, 7 and n = 2, 3, 4 with 100 randomly chosen real data for each case. For all d = 3, 5, 7, the maximum possible numbers seem to be 3, 7, 15 for n = 2, 3, 4, respectively, but we do not have any mathematical proof for them. Conjecture 16. The number of real critical points of (1) is at most 2n − 1.
2
We note that Theorem 15 is also valid for the scaled Fermat hypersurface S F na,d since the critical system contains the same number of monomials for all scaling vectors a. 3. Scaled Fermat hypersurfaces 3.1. Genericity of scaled Fermat hypersurfaces Recall the relation (3)
F n,d = deg(S) − mult(0) − (n, d). The first two terms in this expression are invariant under any G L (n + 1, C) action (acting on the variables), thus we only focus on the last term (n, d), which is a sum of δ(m, d − 2) with binomial coefficients (see Lemma 11). For a given scaling vector a ∈ (C∗ )m+1 , define δ(m, p , a) to be the number of solutions of
2 1 + x21 + · · · + xm =0 p
xi = ai /a0 for each i = 1, · · · , m whose entries are all nonzero. Note that δ(m, p , 1) = δ(m, p ) where 1 = (1, · · · , 1). For I ⊆ {0, . . . , n}, let a I = (ai 0 , . . . , ai | I |−1 ) where ai j is the j-th entry of a. Now the ED-degree of S F na,d is given by
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EDdeg( S F na,d ) = d
n −1
(d − 1)i −
δ(| I | − 1, d − 2, a I ).
I ⊆{0,...,n}
i =0
Therefore the ED-degree of S F na,d achieves the equality in the general bound (Proposition 1) if and only if the latter summands are all zero. To examine, we need to define the exponential cyclotomic polynomial Q m, p ∈ Z[x0 , . . . , xm ]. For an integer p and a primitive p-th root of unity ζ , consider the polynomial p
P m, p ( A 0 , . . . , A m ) =
A0 +
t 1 ,...,tm =1
m
tk
ζ Ai .
k =1 p
p
p
One can easily see that P m, p ( A 0 , . . . , A m ) ∈ Z[ A 0 , . . . , A m ]. Replace A i by xi to get a polynomial Q m, p (x0 , . . . , xm ), i.e., Q m, p is the unique polynomial such that
p Q m, p ( A 0 , . . . ,
p
A m ) = P m, p ( A 0 , . . . , A m ) .
Theorem 17. δ(m, p , a) = 0 if and only if
2 Q m, p (a20 , . . . , am )=0
for p odd,
Q m, p /2 (a0 , . . . , am ) = 0 for p even.
2
Proof. Let a ∈ (C∗ )n+1 be given. We may assume a0 = 1. For each i, choose a complex number b i so p that b i = ai . Let ζ be a primitive p-th root of unity. By definition, δ(m, p , a) = 0 if and only if the system
2 1 + x21 + · · · + xm =0 p xi = ai for each i = 1, · · · , m
has a solution whose entries are all nonzero. Any solution of the second equations is of the form 2 2tm (b1 ζ t1 , . . . , bm ζ tm ). It satisfies the first equation if and only if 1 + b21 ζ 2t1 + · · · + bm ζ = 0. The image 2 of the square map z → z defined on the set of all p-th roots of unity is itself if p is odd, or is the set of all ( p /2)-nd roots of unity if p is even. Therefore δ(m, p , a) = 0 if and only if p
2 tm (1 + b21 ζ t1 + · · · + bm ζ )=0
t 1 ,...,tm =1
for d odd, or p /2
2 (1 + b21 (ζ 2 )t1 + · · · + bm (ζ 2 )tm ) = 0
t 1 ,...,tm =1 p
for d even. Now the theorem follows by replacing b i with ai after expanding the product.
2
Corollary 18. For generic a ∈ (C∗ )n+1 ,
EDdeg( S F na,d ) = d
n −1
(d − 1)i
2
i =0
The exponential cyclotomic polynomial Q m, p would be interesting itself. We close this section with a theorem showing that Q m, p has a nice property as an algebraic object. Theorem 19. For any integer m and p, the exponential cyclotomic polynomial Q m, p is irreducible over C.
2
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p
Proof. Let f ∈ C[x0 , . . . , xm ] be an irreducible factor of Q m, p . Then f ( A 0 , . . . , A m ) is a factor p p of P m, p ( A 0 , . . . , A m ), and we may assume that f ( A 0 , . . . , A m ) is divisible by ( A 0 + · · · + A m ). p p For any (t 1 , . . . , tm ), t i = 1, . . . , p, the polynomial f ( A 0 , . . . , A m ) is invariant under the action p p ti A i → ζ t i A i . Therefore f ( A 0 , . . . , A m ) is divisible by ( A 0 + i ζ A i ). Since (t 1 , . . . , tm ) was arp p bitrary, f ( A 0 , . . . , A m ) is divisible by every possible linear factor of P m, p ( A 0 , . . . , A m ). Therefore p p f ( A 0 , . . . , A m ) = P m, p ( A 0 , . . . , A m ) and hence Q m, p = f (x0 , . . . , xm ) up to scalar multiplication. 2 Acknowledgements The author would like to thank his advisors Bernd Sturmfels and Hyungju Park for their guidance, comments and support, and Donghoon Hyeon for useful conversations and suggestions. Discussions with Donggeon Yhee about exponential cyclotomic polynomials were also helpful. References Adams, S., Sarnak, P., 1994. Betti numbers of congruence groups. Isr. J. Math. 88, 31–72. Agostini, D., Alberelli, D., Grande, F., Lella, P., 2015. The maximum likelihood degree of Fermat hypersurfaces. J. Algebr. Statist. 6, 108–132. Cox, D., Little, J., O’Shea, D., 1992. Ideals, Varieties, and Algorithms. An Introdution to Computational Algebraic Geometry and Commutative Algebra. Undergrad. Texts Math. Springer-Verlag, New York. Draisma, J., Horobet,¸ E., Ottaviani, G., Sturmfels, B., Thomas, R., 2016. The Euclidean distance degree of an algebraic variety. Found. Comput. Math. 16, 99–149. Grayson, D., Stillman, M., 1992. Macaulay2, a software system for research in algebraic geometry, available at http://www. math.uiuc.edu/Macaulay2/. Huh, J., Sturmfels, B., 2014. Likelihood geometry. In: Conca, A., et al. (Eds.), Combinatorial Algebraic Geometry. In: Lect. Notes Math., vol. 2014. Springer, pp. 63–117. Khovanskii, A.G., 1980. On a class of systems of transcendental equations. Dokl. Akad. Nauk SSSR 255 (4), 804–807. Laurent, M., 1984. Equations diophantiennes exponentielles. Invent. Math. 78, 299–327. Mann, H., 1965. On linear relations between roots of unity. Mathematika 12, 107–117. Ottaviani, G., Spaenlehauer, P.-J., Sturmfels, B., 2014. Exact solutions in structured low-rank approximation. SIAM J. Matrix Anal. Appl. 35 (4), 1521–1542.
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Contents lists available at ScienceDirect
Journal of Symbolic Computation www.elsevier.com/locate/jsc
The Euclidean distance degree of Fermat hypersurfaces Hwangrae Lee 1 Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea
a r t i c l e
i n f o
Article history: Received 3 July 2015 Accepted 3 July 2016 Available online xxxx Keywords: Euclidean distance degree Fermat hypersurface Optimization
a b s t r a c t Finding the point in an algebraic variety that is closest to a given point is an optimization problem with many applications. We study the case when the variety is a Fermat hypersurface. Our formula for its Euclidean distance degree is a piecewise polynomial whose pieces are defined by subtle congruence conditions. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Let X ⊂ Rn be a real affine algebraic variety, i.e., X is the common zero set of some polynomials f 1 , . . . , f m ∈ R[x1 , . . . , xn ]. We consider the following problem: given u ∈ Rn , compute u ∗ ∈ X that n minimizes the squared Euclidean distance du (x) = i =1 (u i − xi )2 from the given point u. This problem arises from best approximation problems. Once we observe a data u in a mathematical model X by experiment, the data may be perturbed by some noise so that u does not actually lie on X . In such a situation, the nearest point u ∗ to u in X represents the original data suggested by u. In order to find u ∗ algebraically, we consider the zeroes in Cn of the equations defining X , and n we examine all complex critical points of the squared distance function du (x) = i =1 (u i − xi )2 on X \ X sing where X sing is the singular locus of X . If X has a positive dimensional singular locus, then there could be infinitely many critical points of du (x) on X . Thus we remove the singular locus of X . The number of such critical points is finite and constant on a dense open subset of data u ∈ Rn . That number of critical points was studied by Draisma et al. (2016). It is called the Euclidean distance
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E-mail address: [email protected]. URL: http://myhome.postech.ac.kr/meso/. The author was supported by IBS-R003-D1.
http://dx.doi.org/10.1016/j.jsc.2016.07.006 0747-7171/© 2016 Elsevier Ltd. All rights reserved.
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degree (ED-degree) of the variety X , and is denoted as EDdeg( X ). From now on, all the objects will be considered as complex varieties, except in Section 2.3. Sometimes, X is given by homogeneous polynomials. The set of m1 × m2 matrices of rank at most k is a typical example. Such a variety is called a projective algebraic variety in Pn (C). For the definition of Pn (C) and more informations, see Chapter 8 of the book by Cox et al. (1992). For a projective X ⊂ Pn (C), we define EDdeg( X ) to be the ED-degree of the affine cone of X in Cn+1 . That is, just regard X as an affine variety and compute the ED-degree. The ED-degrees of determinantal varieties as above have been studied by Ottaviani et al. (2014). This paper is motivated by following general upper bound on the ED-degree. Proposition 1 (Draisma et al., 2016). Let X be a hypersurface in Pn (C) defined by a homogeneous polynomial f of degree d. Then
EDdeg( X ) ≤ d
n −1 (d − 1)i , i =0
and equality holds when f is generic.
2
In this paper, we focus on Fermat hypersufaces and their variations. Definition 2.
• A Fermat hypersurface of degree d in Pn (C), denoted by F n,d , is the projective variety defined by the polynomial xd0 + · · · + xnd . • An affine Fermat hypersurface of degree d in Cn , denoted by A F n,d , is the affine variety defined by the polynomial xd1 + · · · + xnd − 1. • A scaled Fermat hypersurface of degree d in Pn (C) with scaling vector a = (a0 , . . . , an ) ∈ (C∗ )n+1 , denoted by S F na,d , is the projective variety defined by the polynomial xd0 /a0 + · · · + xnd /an . 2 In statistical optimization, maximum likelihood estimation (MLE) is an important tool. The generic number of the critical points of the maximum likelihood function, called ML-degree, is a parallel concept to ED-degree. The ML-degrees of many statistically relevant varieties have been computed (Huh and Sturmfels, 2014). Recently, in particular, the ML-degree of F n,d is partially given by Agostini et al. (2015). Their results, which we review in Example 8, serve as motivation to our study of the ED-degree of F n,d . This paper is organized as follows. In Section 2, we will investigate the sharpness of the general bound (Proposition 1) for the Fermat hypersurfaces. We give a formula for the ED-degree of F n,d (Theorem 4), that gives an explicit formula for n ≤ 3 (Remark 5, Example 8). If we fix n and consider the general bound as a function in d, it is the best possible polynomial bound (Lemma 6), while the gap can be arbitrarily large (Remark 5). The main theorem can be used for an efficient algorithm which computes the ED-degree of a Fermat hypersurface numerically (Example 13). The proof of Theorem 4 can be used similarly to evaluate the ED-degree for an affine Fermat hypersurface A F n,d (Corollary 14). After that, an open problem (Conjecture 16) about real Fermat hypersurfaces will be discussed. In Section 3, we will consider the scaled Fermat hypersurfaces for fixed n and d. We introduce the exponential cyclotomic polynomial Q m, p which has a special role for the scaling vector a of S F na,d (Theorem 17). As a corollary, we will see that the ED-degree of a scaled Fermat hypersurface usually achieves the general bound of Proposition 1. 2. ED-degree for Fermat hypersurfaces 2.1. Main theorem for Fermat hypersurfaces In this section, we compute the ED-degree of F n,d for each n, d.
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Definition 3. For a positive integer p, fix a p-th primitive root of unity ζ . Define δ(m, p ) to be the number of integer m-tuples (t 1 , . . . , tm ), 1 ≤ t i ≤ p, satisfying
1+
m
ζ 2t i = 0,
i =1
or equivalently, satisfying that the univariant polynomial 1 + x2t i is a multiple of the p-th cyclotomic polynomial p (x). Note that it does not depend on the choice of ζ . 2 Theorem 4. For d ≥ 3, the ED-degree of the Fermat hypersurface F n,d is given by
EDdeg( F n,d ) = d
n −1 n n+1 · δ(m, d − 2). (d − 1)i − m+1 i =0
2
m =1
Theorem 4 will be proved further below. Remark 5. For small m, the following are derived easily from the definition:
(i)
(ii)
(iii)
δ(1, p ) =
2 if p ≡ 0 mod 4 0 otherwise
⎧ ⎨ 8 if p ≡ 0 mod 6 δ(2, p ) = 2 if p ≡ 3 mod 6 ⎩ 0 otherwise 12p − 24 if p ≡ 0 mod 4 δ(3, p ) = 0
otherwise
In particular, (iii) implies that the difference between the general bound and the actual ED-degree can be arbitrarily large. Although, the following lemma shows that the general bound is the best possible polynomial bound. 2 Lemma 6. If p is a prime bigger than m + 1, then δ(m, p ) = 0.
2
m
Proof. Assume 1 + i =1 ζ 2t i = 0. Replacing 2t i by 2t i − p if 2t i ≥ p for each i, we have a polynomial in ζ whose degree is less than p. Since p is a prime, it should be a scalar multiple of the cyclotomic polynomial p (ζ ) := 1 + ζ + · · · + ζ p −1 . It has p terms, hence m + 1 ≥ p. 2 No closed formula for δ(m, p ) is known, but it has been studied in both algebraic geometry and number theory (Adams and Sarnak, 1994; Laurent, 1984; Mann, 1965). In particular, a theorem given by Laurent (1984, Theorem 2) implies that δ(m, p ) is a polynomial periodic function in p in the sense of Adams and Sarnak (1994). Corollary 7. For fixed n, the ED-degree of F n,d is a polynomial periodic function in d. That is,
EDdeg( F n,d ) = for some periodic c i ’s.
c i (d)di
2
Example 8. Agostini et al. (2015) showed that the ML-degrees of the Fermat curves (n = 2) are given by
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⎧ 2 ⎪ ⎪ d2 + d ⎨ d +d−3 MLdeg( F 2,d ) = ⎪ d2 + d − 2 ⎪ ⎩ 2 d +d−5
if d ≡ 0, 2 mod 6 if d ≡ 3, 5 mod 6 if d ≡ 4 mod 6 if d ≡ 1 mod 6.
By the Theorem 4, we have
⎧ 2 d ⎪ ⎪ ⎪ 2 ⎪ ⎨d −2 EDdeg( F 2,d ) = d2 − 6 ⎪ ⎪ ⎪ d2 − 8 ⎪ ⎩ 2 d − 14
if d ≡ 0, 1, 3, 4, 7, 9 mod 12 if d ≡ 5, 11 mod 12 if d ≡ 6, 10 mod 12 if d ≡ 8 mod 12 if d ≡ 2 mod 12.
It is a polynomial periodic function in d, and the general bound EDdeg ( F 2,d ) ≤ d2 is the best possible polynomial bound. Comparing with MLdeg, both are periodic while their periods are different. 2 The system for critical points of the distance function is given by
xd0 + · · · + xnd = 0
(1)
xdi −1 (x j − u j ) = xdj −1 (xi − u i ) for each i = j
where the vector u = (u 0 , . . . , un ) ∈ Cn is sufficiently generic. The ED-degree is the number of solutions of (1) except (0, . . . , 0), which is a (unique) singular point of the affine cone over the Fermat hypersurface F n,d . Introducing a new variable t, we modify the system (1) into the following homogeneous system in S [t ] = C[x0 , . . . , xn , t ]:
S=
xd0 + · · · + xnd = 0
(2)
xdi −1 (x j − u j t ) = xdj −1 (xi − u i t ) for each i = j .
Each solution of S of the form (c 0 : · · · : cn : 1) corresponds to the solution (c 0 , . . . , cn ) of (1). The system S has more solutions that we do not want to count. Let mult (0) be the multiplicity of (0 : · · · : 0 : 1) for the system S, and (n, d) be the number of solutions of the form (a0 : · · · : an : 0) counting multiplicities. Then the ED-degree of F n,d is given by
EDdeg( F n,d ) = deg(S) − mult(0) − (n, d)
(3)
where deg(S) is the degree of the projective scheme defined by the system S. Now, Theorem 4 is just a consequence of following lemmas. Lemma 9. The multiplicity of (0 : · · · : 0 : 1) for the system S, denoted by mult(0), is d(d − 1)n .
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Proof. Let I be the ideal in S [t ] generated by equations in S and m = (x0 , . . . , xn ) be the ideal corresponding to the point (0 : · · · : 0 : 1). Then mult(0) is defined by the length of S [t ]m / I m as an S [t ]m -module. In the local ring S [t ]m , the factor (xi − u i t ) is a unit. (By the genericity of u i , we may assume u i = 0 for all i.) Writing μi = (x0 − u 0 t ) · (xi − u i t )−1 , the localized ideal I m is generated by
xd0 + · · · + xnd = 0
xd0−1 = xdi −1 μi for each i .
(4)
Here, the length of S [t ]m / I m is just the maximum number of monomials in S which are independent α α modulo I m . By direct counting, we see that mult(0) = |{x0 0 · · · xn n | 0 ≤ α0 ≤ d − 1, 0 ≤ α1 , . . . , αn ≤ n ¯ ¯ d − 2}| = d(d − 1) . Alternatively, it is same as dimC ( S / I ) where I is the ideal in S defined by (4) after changing each μi into arbitrary nonzero value in C. Therefore, by Bézout theorem, we get the same answer. 2
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To compute (n, d) in (3), we want to put t = 0 in the system S to get
xd0 + · · · + xnd = 0
(5)
xdi −1 x j = xdj −1 xi for each i = j
This could give the wrong answer if S and the hyperplane t = 0 meet non-transversally. The next lemma shows that it is not the case. Lemma 10. The system S and the hyperplane t = 0 meet transversally. Hence (n, d) = deg((5)).
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Proof. It suffices to show that the system S and ∂S/∂ t have no common root where ∂S/∂ t is the ideal generated by the t-directional partial derivatives of the equations in S. Let c = (c 0 : · · · : cn : 0) be a nonzero solution of S. It has at least two nonzero entries by the equation cd0 + · · · + cnd = 0. Without loss of generality, we may assume c 0 = 1 and c 1 = 0, which implies cd1−2 = 1. Also,
∂ d −1 x0 (x1 − u 1 t ) − xd1−1 (x0 − u 0 t ) |x=c ∂t = −u 1 + cd1−1 u 0 = 0 only if |u 0 | = |u 1 |.
By the genericity of u i , the last equality does not happen.
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Lemma 11. Let δ(m, p ) be the function defined in Definition 3. Then we have
(n, d) =
n n+1 m =1
m+1
· δ(m, d − 2).
2
Proof. By Lemma 10, (n, d) = deg((5)). Let (c 0 : · · · : cn ) be a solution of (5). Suppose that c i = 0 for all i. Then x0 = 1 in the system (5) implies that all c i ’s are some (d − 2)-nd roots of unity. Fix a (d − 2)-nd primitive root of unity ζ , and write c i = ζit . Then the system (5) has δ(n, d − 2) many solutions. If a solution has m + 1 many nonzero coordinates, the number of such solutions is (for the choices of nonzero coordinates) times δ(m, d − 2). 2
n+1 m+1
Lemma 12. The degree of the system S is given by
deg(S) = d
n
(d − 1)i .
2
i =0
Proof. Let I be the ideal generated by
xdi −1 (x j − u j t ) = xdj −1 (xi − u i t ) for each i = j
= 2 × 2 minors of
xd0−1 · · · xnd−1 x0 − u 0 t · · · xn − un t
It defines a curve in PnC+1 . By Lemma 10, deg( I ) = deg( I + (t )). If c = (c 0 : · · · : cn : 0) is a solution for I + (t ), write cm = 1 where m is the first nonzero entry of c. For each nonzero entry except cm , there are d − 2 choices to be a solution. Hence the total number of solutions is
n +1 n+1 i =1
i
(d − 2)i −1 =
n (d − 1)i . i =0
Therefore deg(S) = deg( F n,d ) · deg( I ) = d
n
i =0 (d
− 1)i by Bézout. 2
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Proof of Theorem 4. We have
EDdeg( F n,d ) = deg(S) − mult(0) − (n, d). Apply Lemma 12, 9, and 11 to each term in the right side.
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Example 13. We showed that the ED-degrees of the Fermat hypersurfaces can be computed by δ(m, p ) or (m, p ) without using the random data u. The following Macaulay2 (Grayson and Stillman, 1992) code computes the ED-degree of F n,d efficiently.
n=2,d=5; R=QQ[x_0..x_n]; gbd=sum apply(n,i->d*(d-1)^i); -- the general bound F=sum apply(n+1,i->(gens R)_i^2); M=matrix{apply(n+1,i->((gens R)_i)^(d-1))}||matrix{gens R}; I=ideal(F)+minors(2,M); EDdeg=gbd-(degree I)*(dim I) The output reveals that the Fermat quintic cone F 2,5 has ED-degree 23.
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2.2. Affine Fermat hypersurfaces Let X be an affine Fermat hypersurface A F n,d . The system for the critical points of the distance function is given by
xd1 + · · · + xnd = 1
xdi −1 (x j − u j ) = xdj −1 (xi − u i ) for each i = j and the homogenized system is
xd1 + · · · + xnd = t d
(6)
xdi −1 (x j − u j t ) = xdj −1 (xi − u i t ) for each i = j
In this case, (0 : · · · : 0 : 1) is not a solution for (6) (see Lemma 9). Except that, the ED-degree of A F n,d can be computed in the same way as in the homogeneous cases. Corollary 14. For d ≥ 3, the ED-degree of the affine Fermat hypersurface A F n,d is given by
EDdeg( A F n,d ) = d
n −1
(d − 1)i −
i =0
n −1 n · δ(m, d − 2). m+1
2
m =1
Note that the first summand is the general bound for affine varieties, given by Draisma et al. (2016, Corollary 2.5). 2.3. Real critical points For odd d, the Fermat hypersurface F n,d can be considered as a nonempty real variety. In this case, the number of the real critical points of the squared distance function highly depends on the location of the given point u ∈ Rn+1 . Nonetheless, the next theorem gives an upper bound for the maximum possible (finite) number of the real critical points. By possible number we mean that there is an open set in Rn+1 of data space such that the number is attained for all points in the open set.
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Theorem 15. For the Fermat hypersurface F n,d , the number of the real critical points of the squared distance function is bounded by
√
2
25n2 −3n+2
· (n + 2)5n
2
Proof. Let u ∈ Rn+1 be a point not in F n,d , whose entries are all nonzero. Then the critical equation (1) can be written by
xd0 + · · · + xnd = 0
xdi −1 (x0 − u 0 ) = xd0−1 (xi − u i ) for each i = 1, . . . , n. This system has n + 1 polynomials in n + 1 variables, and the number of monomials used in this system is 5n + 1, which does not depend on d. By Khovanskii’s fewnomial bound (Khovanskii, 1980), this system has at most
2
5n 2
· (n + 2)5n
positive solutions. It is also an upper bound for the number of real solutions in any orthant, and there is no solution with 0 in its coordinates. Hence we can have at most
2n+1 · 2
5n 2
· (n + 2)5n
real solutions in total.
2
Note that this bound does not depend on d, hence we can ask for the sharp bound for each n. For n = 1, the real cone of F 1,d is a straight line in R2 , hence the critical equation has one real solution. We numerically computed the numbers of real solutions for d = 3, 5, 7 and n = 2, 3, 4 with 100 randomly chosen real data for each case. For all d = 3, 5, 7, the maximum possible numbers seem to be 3, 7, 15 for n = 2, 3, 4, respectively, but we do not have any mathematical proof for them. Conjecture 16. The number of real critical points of (1) is at most 2n − 1.
2
We note that Theorem 15 is also valid for the scaled Fermat hypersurface S F na,d since the critical system contains the same number of monomials for all scaling vectors a. 3. Scaled Fermat hypersurfaces 3.1. Genericity of scaled Fermat hypersurfaces Recall the relation (3)
F n,d = deg(S) − mult(0) − (n, d). The first two terms in this expression are invariant under any G L (n + 1, C) action (acting on the variables), thus we only focus on the last term (n, d), which is a sum of δ(m, d − 2) with binomial coefficients (see Lemma 11). For a given scaling vector a ∈ (C∗ )m+1 , define δ(m, p , a) to be the number of solutions of
2 1 + x21 + · · · + xm =0 p
xi = ai /a0 for each i = 1, · · · , m whose entries are all nonzero. Note that δ(m, p , 1) = δ(m, p ) where 1 = (1, · · · , 1). For I ⊆ {0, . . . , n}, let a I = (ai 0 , . . . , ai | I |−1 ) where ai j is the j-th entry of a. Now the ED-degree of S F na,d is given by
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EDdeg( S F na,d ) = d
n −1
(d − 1)i −
δ(| I | − 1, d − 2, a I ).
I ⊆{0,...,n}
i =0
Therefore the ED-degree of S F na,d achieves the equality in the general bound (Proposition 1) if and only if the latter summands are all zero. To examine, we need to define the exponential cyclotomic polynomial Q m, p ∈ Z[x0 , . . . , xm ]. For an integer p and a primitive p-th root of unity ζ , consider the polynomial p
P m, p ( A 0 , . . . , A m ) =
A0 +
t 1 ,...,tm =1
m
tk
ζ Ai .
k =1 p
p
p
One can easily see that P m, p ( A 0 , . . . , A m ) ∈ Z[ A 0 , . . . , A m ]. Replace A i by xi to get a polynomial Q m, p (x0 , . . . , xm ), i.e., Q m, p is the unique polynomial such that
p Q m, p ( A 0 , . . . ,
p
A m ) = P m, p ( A 0 , . . . , A m ) .
Theorem 17. δ(m, p , a) = 0 if and only if
2 Q m, p (a20 , . . . , am )=0
for p odd,
Q m, p /2 (a0 , . . . , am ) = 0 for p even.
2
Proof. Let a ∈ (C∗ )n+1 be given. We may assume a0 = 1. For each i, choose a complex number b i so p that b i = ai . Let ζ be a primitive p-th root of unity. By definition, δ(m, p , a) = 0 if and only if the system
2 1 + x21 + · · · + xm =0 p xi = ai for each i = 1, · · · , m
has a solution whose entries are all nonzero. Any solution of the second equations is of the form 2 2tm (b1 ζ t1 , . . . , bm ζ tm ). It satisfies the first equation if and only if 1 + b21 ζ 2t1 + · · · + bm ζ = 0. The image 2 of the square map z → z defined on the set of all p-th roots of unity is itself if p is odd, or is the set of all ( p /2)-nd roots of unity if p is even. Therefore δ(m, p , a) = 0 if and only if p
2 tm (1 + b21 ζ t1 + · · · + bm ζ )=0
t 1 ,...,tm =1
for d odd, or p /2
2 (1 + b21 (ζ 2 )t1 + · · · + bm (ζ 2 )tm ) = 0
t 1 ,...,tm =1 p
for d even. Now the theorem follows by replacing b i with ai after expanding the product.
2
Corollary 18. For generic a ∈ (C∗ )n+1 ,
EDdeg( S F na,d ) = d
n −1
(d − 1)i
2
i =0
The exponential cyclotomic polynomial Q m, p would be interesting itself. We close this section with a theorem showing that Q m, p has a nice property as an algebraic object. Theorem 19. For any integer m and p, the exponential cyclotomic polynomial Q m, p is irreducible over C.
2
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p
Proof. Let f ∈ C[x0 , . . . , xm ] be an irreducible factor of Q m, p . Then f ( A 0 , . . . , A m ) is a factor p p of P m, p ( A 0 , . . . , A m ), and we may assume that f ( A 0 , . . . , A m ) is divisible by ( A 0 + · · · + A m ). p p For any (t 1 , . . . , tm ), t i = 1, . . . , p, the polynomial f ( A 0 , . . . , A m ) is invariant under the action p p ti A i → ζ t i A i . Therefore f ( A 0 , . . . , A m ) is divisible by ( A 0 + i ζ A i ). Since (t 1 , . . . , tm ) was arp p bitrary, f ( A 0 , . . . , A m ) is divisible by every possible linear factor of P m, p ( A 0 , . . . , A m ). Therefore p p f ( A 0 , . . . , A m ) = P m, p ( A 0 , . . . , A m ) and hence Q m, p = f (x0 , . . . , xm ) up to scalar multiplication. 2 Acknowledgements The author would like to thank his advisors Bernd Sturmfels and Hyungju Park for their guidance, comments and support, and Donghoon Hyeon for useful conversations and suggestions. Discussions with Donggeon Yhee about exponential cyclotomic polynomials were also helpful. References Adams, S., Sarnak, P., 1994. Betti numbers of congruence groups. Isr. J. Math. 88, 31–72. Agostini, D., Alberelli, D., Grande, F., Lella, P., 2015. The maximum likelihood degree of Fermat hypersurfaces. J. Algebr. Statist. 6, 108–132. Cox, D., Little, J., O’Shea, D., 1992. Ideals, Varieties, and Algorithms. An Introdution to Computational Algebraic Geometry and Commutative Algebra. Undergrad. Texts Math. Springer-Verlag, New York. Draisma, J., Horobet,¸ E., Ottaviani, G., Sturmfels, B., Thomas, R., 2016. The Euclidean distance degree of an algebraic variety. Found. Comput. Math. 16, 99–149. Grayson, D., Stillman, M., 1992. Macaulay2, a software system for research in algebraic geometry, available at http://www. math.uiuc.edu/Macaulay2/. Huh, J., Sturmfels, B., 2014. Likelihood geometry. In: Conca, A., et al. (Eds.), Combinatorial Algebraic Geometry. In: Lect. Notes Math., vol. 2014. Springer, pp. 63–117. Khovanskii, A.G., 1980. On a class of systems of transcendental equations. Dokl. Akad. Nauk SSSR 255 (4), 804–807. Laurent, M., 1984. Equations diophantiennes exponentielles. Invent. Math. 78, 299–327. Mann, H., 1965. On linear relations between roots of unity. Mathematika 12, 107–117. Ottaviani, G., Spaenlehauer, P.-J., Sturmfels, B., 2014. Exact solutions in structured low-rank approximation. SIAM J. Matrix Anal. Appl. 35 (4), 1521–1542.