Gröbner bases and their application to the Cauchy problem on finitely generated affine monoids

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Gröbner bases and their application to the Cauchy problem on finitely generated affine monoids Martin Scheicher 1 Dipartimento di Ingegneria dell’Informazione, Università di Padova, via Gradenigo 6/B, 35131 Padova, Italy

a r t i c l e

i n f o

Article history: Received 29 October 2015 Accepted 30 May 2016 Available online xxxx MSC: 13P10 39A06 Keywords: Gröbner basis Generalised term order Cauchy problem for partial difference equations

a b s t r a c t For finitely generated submonoids of the integer lattice and submodules over the associated monoid algebra, we investigate Gröbner bases with respect to generalised term orders. Up to now, this theory suffered two disadvantages: The algorithm for computing the Gröbner bases was slow and it was not known whether there existed generalised term orders for arbitrary finitely generated submonoids. This limited the applicability of the theory. Here, we describe an algorithm which transports the problem of computing the Gröbner bases to one over a polynomial ring and use the conventional Gröbner theory to solve it, thus making it possible to apply known, optimised algorithms to it. Furthermore, we construct generalised term orders for arbitrary finitely generated submonoids. As an application we solve the Cauchy problem (initial value problem) for systems of linear partial difference equations over finitely generated submonoids. © 2016 Published by Elsevier Ltd.

1. Introduction Let F be a field and let N be a finitely generated submonoid of Z1×n . We consider the system of partial linear difference equations

1

E-mail address: [email protected]. The author has been supported by the Austrian FWF via project J3411-N25.

http://dx.doi.org/10.1016/j.jsc.2016.07.002 0747-7171/© 2016 Published by Elsevier Ltd.

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l  

R i j (μ) w j (μ + η) = v i (η),

where i ∈ {1, . . . , k} and η ∈ N ,

(1)

j =1 μ∈ N

with constant coefficients R i j (μ) ∈ F (only finitely many of them nonzero), inhomogeneity v ∈ ( F N )k , and solution w ∈ ( F N )l . The Cauchy problem for this kind of system is the task of finding suitable subsets in {1, . . . , l} × N which can be interpreted as initial value areas2 as well as a method for computing the solution from its initial values and the inhomogeneity v. Difference equations over N = Z1×n or N = N1×n1 × Z1×n2 and their initial value problem are a standard subject in multidimensional systems theory and signal processing, see, for example, Lim (1988) and Park and Regensburger (2007). More general monoids appear in connection with periodic difference equations (Bourlès et al., 2015). In order to be able to use algebraic techniques, we reformulate the system (1) by introducing the monoid algebra F [ N ] of N, which is a subalgebra of the algebra F [Z1×n ] = F [σ , σ −1 ] = F [σ1 , . . . , σn , σ1−1 , . . . , σn−1 ] of Laurent polynomials in n variables. The elements of F [ N ] act on the N sequence space F N by shifts, i.e., (σ μ ◦ w )(η) = w (μ + η) for μ, η ∈ N and  w ∈ F . In this way, we can interpret the coefficients in (1) as a matrix R ∈ F [ N ]k×l , where R i j = μ∈ N R i j (μ)σ μ , and write (1) in the compact form R ◦ w = v. The reason why we use the monoid algebra F [ N ] and not just interpret R as a matrix over F [σ , σ −1 ] or F [σ ] (if N ⊆ N1×n ) is that F [ N ] contains information about the structure of the domain of definition N of the solutions w as well as on which shifts are allowed and which are not; and this information is lost if one just considers the Laurent polynomial algebra or the polynomial ring.3 The first instance where Gröbner bases were used for solving the Cauchy problem for systems of difference equations was in Chapter 5 of Oberst’s comprehensive article Oberst (1990). Oberst considered only the submonoid N = N1×n . Since the monoid algebra F [N1×n ] is just a polynomial ring in n variables, the usual theory of Gröbner bases was sufficient. Zerz and Oberst (1993) treated the case that N = Z1×n is the whole integer grid. In this situation, F [ N ] is a Laurent polynomial ring which the authors parametrised by a polynomial ring F [s]. In both papers, the algorithms for computing the values of the solution are recursive methods Oberst (1990, Thm. 63 on p. 98), Zerz and Oberst (1993, Thm. 4 on p. 266). Independently, Zampieri (1994) derived a method for computing the solution of systems on N = Z1×n via the normal forms of monomials in F [s]. Later, Pauer and Zampieri (1996) introduced conic decompositions of arbitrary finitely generated submonoids N and generalised term orders with respect to those decompositions. This enabled them to define and compute Gröbner bases for ideals in F [ N ] and they applied this to the modelling problem. Pauer and Unterkircher (1999) introduced generalised term orders on [l] × N, they defined Gröbner bases for submodules of F [ N ]1×l , gave an algorithm for their computation, and applied this to the homogeneous Cauchy problem (v = 0). Oberst and Pauer (2001, Sec. 2.1) summarised these results using a simpler notation, generalised them to include arbitrary inhomogeneities v and gave new and easier proofs. The method of Zampieri (1994) and Pauer and Unterkircher (1999) for computing Gröbner bases is a generalisation of the Buchberger algorithm which works directly with elements in the monoid al-

2 These areas have the property that, if the system is solvable at all, then any choice of values in the initial value area can be continued uniquely to a solution of the system (1). 3 For example, consider the monoid N = N2 + N3 = {0, 2, 3, 4, 5, . . . } = N \ {1} of N and the matrix R = σ 2 ∈ F [ N ]1×1 , which

gives rise to the difference equation w (μ + 2) = 0 for μ ∈ N, where w = ( w (0), w (2), w (3), w (4), w (5), . . . ) ∈ F N . The solution set of this equation comprises the sequences w where w (0) and w (3) are arbitrary and all the other entries of w are zero. The ideal U = F [ N ] R = F σ 2 + F σ 4 + F σ 5 + F σ 6 + · · · is the set of those linear equations which are satisfied by all solutions of R ◦ w = 0. If we see R only as an element of the polynomial ring F [σ ] and not of the monoid algebra F [ N ], then we consider the ideal U  = F [σ ] R = F σ 2 + F σ 3 + F σ 4 + F σ 5 + · · · instead of U . In this example, the identity U  ∩ F [ N ] = U  holds, thus we have that σ 3 is in U  ∩ F [ N ] but not in U . Thus, U is a proper subset of U  , i.e., U  ∩ F [ N ] does not contain the correct information on the solutions of R ◦ w = 0. Translated into the terms of difference equations, σ 3 ∈ U  ∩ F [ N ] gives rise to the equation σ 3 ◦ w = 0, which means, in particular, that w (3) must be zero. But this equation is not satisfied by all solutions of R ◦ w = 0. Therefore, if we consider U  , we obtain only a proper subset of the solution set of R ◦ w = 0.

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gebra F [ N ]. To use this algorithm, it is necessary to implement it from scratch. Therefore, it is rather slow compared to the current optimised implementations of the Buchberger algorithm for Gröbner bases over the polynomial ring. In addition, while in both articles the authors give examples of generalised term orders, it is not proven that these orders exist for every finitely generated submonoid N and this limits the possible applications of the results. In Section 4, I propose an algorithm for computing Gröbner bases for submodules of F [ N ]1×l with respect to arbitrary generalised term orders using parametrisations of the submonoid N and the monoid algebra F [ N ] by a positive orthant N1×m and a polynomial ring F [s] = F [s1 , . . . , sm ], respectively, as well as the Gröbner bases for submodules over the polynomial ring F [s]. On the parameter space F [s]1×l I use term orders which are compatible with the generalised term order on F [ N ]1×l . In this manner, the main computational task is the calculation of Gröbner bases on F [s]1×l — for which optimised algorithms exist—and thus I work around the computational disadvantage of the algorithms of Zampieri (1994) and Pauer and Unterkircher (1999). In Section 6, I show how conic decompositions and generalised term orders with respect to them can be derived for any finitely generated submonoid N. In Section 5, I give a self-contained new proof of the Theorem (Oberst and Pauer, 2001, Thm. 5) on the solution of the Cauchy problem. Section 7 consists of detailed examples and the specialisations to the case where the submonoid N is the integer lattice Z1×n . 2. Gröbner bases over finitely generated submonoids In the following, we establish the notation we use in this article and recapitulate the definition of a Gröbner basis according to Pauer and Unterkircher (1999, Sec. 2) and Oberst and Pauer (2001, Eq. (25) on p. 267). Let N ⊆ Z1×n be a finitely generated submonoid of the integer grid Z1×n and denote by

ZN =





θ θ; θ ⊆ N finite, μ θ ∈ Z μ

ϑ∈θ

= N − N = {μ − ν ; μ, ν ∈ N} ⊆ Z1×n the abelian group generated by N. The submonoid N gives rise to the monoid algebra

F [ N ] :=



F σ μ ⊆ F [Z1×n ] = F [σ , σ −1 ] = F [σ1 , . . . , σn , σ1−1 , . . . , σn−1 ],

μ∈ N

which is asubalgebra of the Laurent polynomial algebra F [σ , σ −1 ]. The elements of F [ N ] are of the form p = μ∈ N p μ σ μ , where the coefficients p μ lie in F and only finitely many of them are nonzero. For l ∈ N, we use the short hand notation [l] := {1, . . . , l} and the standard basis i

δi := (0, . . . , 0, 1, 0, . . . , 0), of the F [ N ]-module

F [ N ]1×l =



i ∈ [l],



F σ μ δi  p =

(i ,μ)∈[l]× N

p (i ,μ) σ μ δi .

(i ,μ)∈[l]× N

Again, only finitely many coefficients p (i ,μ) ∈ F are nonzero. For arbitrary index sets I , we employ the following general notation. We denote by



F I := w = ( w i )i ∈ I : I −→ F F

(I )



I



and

:= w ∈ F ; supp( w ) finite ⊆ F I

the F -vector spaces of sequences and of finite sequences indexed by I , where supp( w ) is the support

supp( w ) := {i ∈ I ; w i = 0} for w = ( w i )i ∈ I ∈ F I . We use this to identify elements of F [ N ] with the sequences of their coefficients, i.e.,

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F [N ] = F (N )





p μ σ μ = ( p μ )μ∈ N ,

μ∈ N

and in the same manner for F [ N ]1×l ,

F [ N ]1×l =



F σ μ δ i = F ([l]× N ) ,

(i ,μ)∈[l]× N

( p 1 , . . . , pl ) =



p (i ,μ) σ μ δi = ( p (i ,μ) )(i ,μ)∈[l]× N ,

(i ,μ)∈[l]× N

where

pi =



p (i ,μ) σ μ ∈ F [ N ].

μ∈ N

Starting with Equation (4), we will also make the following identification. For a subset J of I , we consider the sequence space F J as a subspace of F I , in detail, id.

F J = { w ∈ F I ; ∀ i ∈ I \ J : w i = 0} ⊆ F I , i.e., we extend the sequences in F J by zero to sequences in F I . In particular, we have that F ( J ) = F J ∩ F (I ) ⊆ F I . Since term orders, i.e., total group orders4 with zero as minimal element, are necessary for a Gröbner bases theory and, if the monoid N is not pointed,5 there exist no term orders on N, we use generalised term orders with respect to conic decompositions of the monoid as in Pauer and Zampieri (1996, Def. 2.1, Def. 2.2) and Pauer and Unterkircher (1999, Def. 2.2, Def. 2.3). A conic decomposition of

N is a finite family ( N J ) J ∈J of finitely generated pointed submonoids N J with Z N J = Z N such that J ∈J N J = N. A generalised term order  on [l] × N with respect to the





conic decomposition ( N J ) J ∈J is a total order which satisfies that (i , 0) = min {i } × N for all i ∈ [l] as well as

∀ J ∈ J ∀(i , μ) ∈ [l] × N ∀( j , ν ) ∈ [l] × N J ∀η ∈ N J :  (i , μ)  ( j , ν ) =⇒ (i , μ + η)  ( j , ν + η) .

(2)

Thus, if restricted to any single N J , J ∈ J , a generalised term order is a term order. When several N J are involved, however, the group order property is sacrificed in favour of the well-orderedness, i.e., the property that zero is minimal. A generalised term order allows us to define the degree

deg( p ) := max supp( p ) ∈ [l] × N of p ∈ F [ N ]1×l \ {0} as well as the degree set of a subset X ⊆ F [ N ]1×l , namely





deg( X ) = deg( p ); p ∈ X \ {0} ⊆ [l] × N . For p ∈ F [ N ]1×l \ {0} let J ∈ J be such that deg( p ) ∈ [l] × N J and let

deg( p σ η ) = deg( p ) + η,

η ∈ N J . Then we have that (3)

where we use the notation

(i , μ) + ν := (i , μ + ν ) for i ∈ [l] and μ, ν ∈ N . 4 A group order is an order where μ  ν implies μ + η  ν + η for μ, ν , η ∈ N. By Kreuzer and Robbiano (2000, Prop. 1.4.14), any total group order on N can be extended uniquely to the group Z N, hence we use the name “group order” even though N is not a group in general. 5 A monoid N is pointed if μ ∈ N and −μ ∈ N implies that μ = 0. An equivalent formulation is that the group of invertible elements of N, namely U( N ) = {μ ∈ N ; −μ ∈ N }, is zero.

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For any submodule U ⊆ F [ N ]1×l there is the direct sum decomposition of F -vector spaces 







F [ N ]1×l = F [l]× N = F ([l]× N )\deg(U ) ⊕ U ,

p = p nf + ( p − p nf ).

(4)

1×l

1×l

The decomposition signifies that for every p ∈ F [ N ] there is a unique p nf ∈ F [ N ] , the normal form6 of p, such that p − p nf ∈ U and supp( p nf ) ∩ deg(U ) = ∅. An exemplary consequence of this direct sum decomposition is the well-known solution of the ideal membership or submodule membership problem, namely p ∈ U if and only if p nf = 0, which is also valid in this generalised setting. According to Pauer and Unterkircher (1999, Def. 2.4), a generating system G of U which satisfies

deg(U ) =



deg( g σ η );



η∈NJ =

g ∈G , J ∈J , deg( g )∈[l]× N J



deg( g ) + N J



(5)

g ∈G , J ∈J , deg( g )∈[l]× N J

is called a Gröbner basis of U . A Gröbner basis G is reduced if, for every g ∈ G , the leading coefficient of g is equal to 1 and the identity

supp( g ) ∩ deg







F [N ] g  = ∅

g  ∈G \{ g }

μ holds. As a consequence, every element g of a reduced Gröbner basis G is of the form g = s δi − μ (s δi )nf , where (i , μ) = deg( g ). In particular, we have that supp( g ) ∩ deg(U ) = deg( g ) . In the case that N = N1×n is the positive orthant, i.e., F [ N ] = F [σ1 , . . . , σn ] is a polynomial ring, and the conic decomposition is trivial, i.e., J has only one element, a generalised term order is a term order and the notions of the degree and of a (reduced) Gröbner basis coincide with the usual ones.

3. Parameter space In this section we parametrise the submonoid N. In Section 3.1 we assume a term order on the parameter space and show some implications. Section 3.2 is devoted to algorithmic aspects. Let θ ⊆ Z1×n be a finite generating system of our submonoid N ⊆ Z1×n , i.e., N = Nθ := ϑ∈θ Nϑ . Then N is parametrised via the surjective linear map

ψ = ·θ : N1×θ −→ N ,

 = ( θ = μ μϑ )ϑ∈θ −→ ψ( μ) = μ



ϑ ϑ. μ

(6)

ϑ∈θ

The map ψ induces the algebra epimorphism  θ μ) = σ μ ϕ : F [s] = F [sϑ ; ϑ ∈ θ] −→ F [ N ], sμ −→ σ ψ( .

Its kernel is the lattice ideal

I N := ker(ϕ ) =







 ν , F sμ − s

(7)

(8)

, μ ν ∈N1×m , ψ( μ)=ψ( ν)

which encodes all the relations of the elements of θ over N, see Miller and Sturmfels (2005, Def. 7.2, Thm. 7.3 on p. 130). Via the homomorphism theorem, we obtain the isomorphism ∼ =

μ) .  −→ σ ψ( ϕind : F [s]/ I N −→ F [ N ], sμ

We extend the maps ψ and

ψ:

ϕ: 6

ϕ , and consequently also ϕind to

[l] × N1×θ −→ [l] × N , F [s]

1×l

−→ F [ N ]

1×l

,

) − (i , μ  → (i , ψ( μ)), and  p = p 1 , . . . , pl −→ ϕ ( p 1 ), . . . , ϕ ( pl ) 

The normal form is sometimes called (tail-) reduced normal form.

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); μ  ∈ ψ −1 (μ)} for and use the same designations for the new maps. Notice that ψ −1 (i , μ) = {(i , μ all μ ∈ N as well as that the kernel of the new map ϕ is I 1N×l = ( I N )1×l . Therefore, we have the isomorphism ∼ =

ϕind : F [s]1×l / I 1N×l −→ F [ N ]1×l . For  p ∈ F [s]1×l the inclusion supp









ϕ ( p ) ⊆ ψ supp( p ) holds.



) ∈ [l] × N1×θ with ψ −1 ψ(i , μ ) ∩ supp( )}. Then ψ(i , μ ) ∈ Lemma 1. Let  p ∈ F [s]1×l and (i , μ p ) = {(i , μ

supp ϕ ( p) . 

Proof. With  p=



 ν p ( j , ν ) s δ j we have that ( j , ν )∈supp( p) 



ϕ ( p) =



ψ( ν)δ =  p ( j , ν)σ j

Because of the assumption, the coefficient of

 

⎞



⎝

( j ,ν )∈[l]× N

( j , ν )∈supp( p)

(i , ν )∈ψ −1

⎛

ν  p ( j , ν)⎠ σ δ j .

( j , ν )∈ψ −1 ( j ,ν )∩supp( p)

μ) δ is σ ψ( i

 p (i , p (i , ν) =  μ)



ψ(i , μ) ∩supp( p)

) ∈ supp( ) ∈ supp and  p (i , p ). Therefore, ψ(i , μ μ) = 0 since (i , μ





ϕ ( p) . 2

3.1. Term orders on the parameter space From now on, we assume a term order  on [l] × N1×θ . The surjective map ψ and the total order  induce the map





ρ : [l] × N −→ [l] × N1×θ , (i , μ) −→ ρ (i , μ) := i , ρ2 (i , μ) := min ψ −1 (i , μ).

(9)

The map ρ is a section of ψ , i.e., the identity ψ ◦ ρ = id[l]× N holds. With  := im(ρ ) = ρ ([l] × N ) we obtain the bijection

ψ| :  ←→ [l] × N : ρ ,

) −→ ψ(i , μ ), (i , μ

(10)

ρ (i , μ) ←−[ (i , μ). 







Lemma 2. Let  p ∈ F [s]1×l . Assume that deg( p ) = ρ ψ(deg( p )) . Then ψ deg( p ) ∈ supp





ϕ ( p) .

Proof. Since deg( p ) is simultaneously the maximum of one set, namely of supp( p ), and the minimum  of another one, namely of ψ −1 ψ(deg( p )) , the intersection of these two sets consists of exactly one element, namely of deg( p ). The assertion follows from Lemma 1. 2





Lemma 3. The identity  = [l] × N1×θ \ deg( I 1N×l ) holds.



Proof. We will show the equality of the complements of these sets, i.e., that the identity [l] × N1×θ ) \

 = deg( I 1N×l ) holds.

   ρ (ψ(i ,  μ)) δ ) ∈ [l] ×N1×θ ) \ . This implies that ρ ψ(i , μ ) ≺ (i , μ ). Therefore,  ⊆. Let (i , μ p := sμ −s 2 i   ). Furthermore, from ψ(i , μ ) = ψ ρ ψ(i , μ ) follows is nonzero and we have that deg( p ) = (i , μ ) = deg( that ϕ ( p ) = 0, i.e.,  p ∈ I 1N×l , and we conclude that (i , μ p ) ∈ deg( I 1N×l ).

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) ∈ deg( I 1N×l ) and let  ). Assume that (i , μ ) ∈  . Then deg( ⊇. Let (i , μ p ∈ I 1N×l with deg( p ) = (i , μ p) =   1×l     (i , μ) = ρ ψ(i , μ) . Lemma 2 implies that ψ(i , μ) ∈ supp ϕ ( p ) . But since p ∈ I N = ker(ϕ )1×l  ) ∈ we have that supp ϕ ( p ) = supp(0) = ∅, which is a contradiction. Consequently, (i , μ  [l] × N1×θ \  . 2 The map

ρ induces the F -linear map

χ : F [ N ]1×l −→ F [s]1×l , σ μ δi −→ sρ2 (i,μ) δi .



 μ This map χ is a section of ϕ . Its image is the space F () = (i , μ)∈ F s δi of polynomials with support in  and we obtain the mutually inverse isomorphisms of F -vector spaces

  μ) δ , sμ δi −→ ϕ (sμ δi ) = σ ψ( i

ϕ | F () : F () ←→ F [ N ]1×l : χ ,

(11)

sρ2 (i ,μ) δi = χ (σ μ δi ) ←−[ σ μ δi . Lemma 4. With respect to the module I 1N×l and the term order , the normal form of  p ∈ F [s]1×l is χ





ϕ ( p) .

 μ

μ)) δ . In particular, the normal form of the monomial s δi is sρ2 (ψ(i , i

Proof. Because of Equation (11) and Lemma 3 we have that





χ ϕ ( p ) ∈ F () = F



([l]× N )\deg( I 1N×l )



.

     Furthermore, ϕ  p − χ ϕ ( p ) = ϕ ( p ) − ϕ χ ϕ ( p ) = ϕ ( p ) − ϕ ( p ) = 0, i.e.,  p − χ ϕ ( p ) ∈ ker(ϕ ) = I 1N×l . The assertion follows because these two properties characterise the normal form uniquely, see Equation (4). 2 3.2. The computation of the lattice ideal In the following, we assume that a finite generating set θ ⊆ Z1×n of N and, if necessary, a term order  on [l] × N1×θ are given. We formulate algorithms to

• compute a generating system of the lattice ideal I N ,  ∈ • decide if μ ∈ Z1×n lies in N and find a preimage μ ψ −1 (μ), • decide if a vector of Laurent polynomials p = (i ,μ)∈[l]×Z1×n p (i ,μ) σ μ δi ∈ F [σ , σ −1 ]1×l = F [Z1×n ]1×l lies in F [ N ]1×l and find a preimage  p ∈ ϕ −1 ( p ), • calculate ρ (i , μ) for (i , μ) ∈ [l] × N and χ ( p ) for p ∈ F [ N ]1×l . There are several algorithms to compute the lattice ideal I N ; see, for example, Theis (1999, Ch. 3) for a nice overview. Although the method we describe here is not the fastest one, it has the decisive advantage that it produces additional data which enable us to solve the remaining three tasks. Our method for the first item is analogous to the one of Conti and Traverso (1991, pp. 131–132), which has its theoretical foundation in Shannon and Sweedler (1988, pp. 267–268) and which has been sketched also in Sturmfels (1996, Alg 4.5, Alg. 5.6). The difference to Conti and Traverso (1991) is that they use the parametrisation of Z1×n from Section 7.2 (which is computationally more effective), while we use the one from Section 7.1, because it is in my opinion more intuitive. The submonoid membership problem (second item) is equivalent to finding a feasible solution of an integer linear programme—see Conti and Traverso (1991, p. 131)—and the computation of ρ (i , μ) can be translated to finding a special optimal solution of such a programme. m×n is a matrix and that the For the algorithmic m purposes of this section we assume that θ ∈ Z submodule N = i =1 Nθi − is generated by the rows θi − , i ∈ [m], of θ . Consequently, N1×m plays the role of N1×θ and the polynomial ring F [s] is F [s] = F [s1 , . . . , sm ]. We consider the parametrisation

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: N1×(2n+m) −→ Z1×n , (μ, ν ,  η) −→ (μ, ν ,  η)   idn with  = − idn ∈ Z(2n+m)×n , θ

of the integer grid Z

1×n

. It induces the algebra homomorphism

 : F [σ , τ , s] −→ F [Z1×n ] = F [σ , σ −1 ],

σ μ τ ν sη −→ σ (μ,τ ,η) , i.e., σi −→ σi , τi −→ σi−1 , si −→ σ θi− , where F [σ , τ , s] is short hand for F [σ1 , . . . , σn , τ1 , . . . , τn , s1 , . . . , sm ]. Define the ideal

J :=

n 

F [σ , τ , s](σi τi − 1) +

i =1



m 

+

−

F [σ , τ , s](si − σ θi− τ θi− ) ⊆ F [σ , τ , s],

i =1



where θ + := max{θi j , 0} (i , j )∈[m]×[n] ∈ Nm×n





and θ − := max{−θi j , 0} (i , j )∈[m]×[n] ∈ Nm×n with θ = θ + − θ − . The first group of generators of J represents the structure of the integer lattice Z1×n while the second one describes the parametrisation ψ of the monoid N. Lemma 5. The ideal J is the lattice ideal of the integer grid Z1×n with respect to the parametrisation . Proof. Clearly, we have that J ⊆ ker() and therefore the induced map

ind : F [σ , τ , s]/ J −→ F [σ , σ −1 ],  p + J −→ ( p ), is well-defined, and it is an epimorphism because  is an epimorphism. In order to prove the assertion, we have to show that J = ker(), i.e., that ind is an isomorphism. For σi + J , τi + J ∈ F [σ , τ , s]/ J we have that

(σi + J )(τi + J ) = σi τi + J = 1 + J ,

i.e., that (σi + J )−1 = τi + J .

This implies that the map

 : F [σi , σi−1 ] −→ F [σ , τ , s]/ J ,

σ μ −→ (σ + J )μ =

n  + − + − (σi + J )μi = (σ + J )μ (τ + J )μ = σ μ τ μ + J , i =1

is well-defined and makes F [σ , τ , s] an F [σ , σ −1 ]-algebra. The identities



ind ◦  (σi ) = ind (σi + J ) = (σi ) = σi and   ind ◦  (σi−1 ) = ind (σi + J )−1 = ind (τi + J ) = (τi ) = σi−1

signify that (ind ◦ ) = id F [σ ,σ −1 ] , i.e., that  is a section of ind and, in particular, injective. The fact that  is an F [σ , σ −1 ]-algebra homomorphism implies that +

−

(σ θi− ) = σ θi− τ θi− + J = si + J

and

ηθ ) = s η + J. (σ 

Thus all the generators of F [σ , τ , s]/ J lie in the image of , i.e.,  is surjective. Since it is a section of ind , it is an isomorphism and its inverse is −1 = ind . Therefore, ind is an isomorphism too. 2 Algorithm 6. Let  be an elimination term order for the σi and τi on F [σ , τ , s], i.e., a term order where any monomial which contains any of the σi or τi is larger than every monomial featuring only the si . From item 3 onward we also use the term order  on [l] × N1×m .

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1. To obtain a generating system of the lattice ideal I N , compute a Gröbner basis J of J with respect to . Then I N is generated by J ∩ F [s]. +

−

2. Let μ ∈ Z1×n . To decide if μ ∈ N and to obtain a preimage, compute the normal form of σ μ τ μ with respect to J and . This normal form is again a monomial. If any of the σi or τi ap / N. Otherwise, the normal form is a monomial sμ with pear therein, then ψ −1 (μ) = ∅, i.e., μ ∈ ψ( μ) = μ .  with ψ( μ) = μ using item 2 and then compute 3. Let (i , μ) ∈ [l] × N. To compute ρ (i , μ), find μ  the normal form of sμ δi with respect to I 1N×l and . According to Lemma 4, this normal form is ν δ with (i ,  s i  ν ) = ρ ( i , μ) . 4. Let p = (i ,μ)∈[l]×Z1×n p (i ,μ) σ μ δi ∈ F [Z1×n ]1×l . To check whether p lies in F [ N ]1×l and to compute a preimage under ϕ , use the procedure from item 2 on all the monomials of p. To compute χ ( p ), feed the result monomial-wise through the algorithm from item 3. Proof. 1. The definition of  implies that | F [s] = ϕ and thus Lem. 5

I N = ker(ϕ ) = ker(| F [s] ) = ker() ∩ F [s] =

J ∩ F [s].

Because of our term order , the elimination theorem—see, for example, Kreuzer and Robbiano (2000, Thm. 3.4.5, item b)—implies that I N = J ∩ F [s] is generated by J ∩ F [s]. ) := min −1 (μ). Since (μ+ , μ− , 0) = μ, Lemma 4 implies that the normal form of 2. Let (ξ, ζ, μ + μ−  μ σ τ with respect to the lattice ideal J is the monomial σ ξ τ ζ sμ . If ξ = ζ = 0 then μ = ) = μ θ = ψ(

(0, 0, μ μ) and consequently μ ∈ N. On the other hand, consider the case that ξi = 0 or ζi = 0 for some i ∈ [n]. Assume that μ ∈ N. Then there is a  ν ∈ N1×m with ψ( ν ) = μ and we have that μ =  ν θ = (0, 0,  ν ). From the fact ) > (0, 0,  that  is an elimination term order for the σi and τi follows that (ξ, ζ, μ ν ) and this ) as the minimal preimage of μ under . Consequently, conflicts with the definition of (ξ, ζ, μ μ ∈/ N. 3. This follows from item 2 and Lemma 4. 4. This follows from items 2 and 3 and the definition of χ . 2 The algorithms described in items 3 and 4 can be improved significantly. Algorithm 7. Let  be an elimination term order for the which satisfies

σi and τi on F [σ , τ , s]1×l , i.e., a term order

   ∀i , j ∈ [n] ∀(ξ, ζ ) ∈ N1×2n \ (0, 0) ∀ η, γ ∈ N1×m : i , (ξ, ζ,  η) > j , (0, 0, γ) .

Let p =



μ (i ,μ)∈[l]×Z1×n p (i ,μ) σ δi

∈ F [Z1×n ]1×l .

1. To test if p ∈ F [ N ]1×l , compute the normal form  p of



p (i ,μ) σ μ

+

−

τ μ δi ∈ F [σ , τ , s]1×l

(i ,μ)∈[l]×Z1×n

with respect to J 1×l and . We have that  p ∈ F [s]1×l if and only if p ∈ F [ N ]1×l and, if this is the case, then  p ∈ ϕ −1 ( p ). 2. To compute χ ( p ), let  be such that the additional condition

    ∀(i ,  η), ( j , γ) ∈ [l] × N1×m : (i ,  η)  ( j , γ) ⇐⇒ i , (0, 0,  η)  j , (0, 0, γ)

is also satisfied. This means that if we identify

[l] × {0}1×2n × N1×m = [l] × N1×m , id.

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then the restriction of  to [l] × {0}1×2n × N1×m is just the given term order  from the beginning of Section 3.1. Apply item 1 to this situation. If it turns out that p ∈ F [ N ]1×l , then the resulting  p is equal to χ ( p ). For (i , μ) ∈ [l] × Z1×n , set p = σ μ δi and apply item 1 to p = σ μ δi to find a preimage of (i , μ). Similarly, item 2 can be used to compute ρ (i , μ). 4. The computation of Gröbner bases In this section we will show in Theorem 11 how to derive Gröbner bases of a submodule U ⊆ F [ N ]1×l via computations in the parameter space F [s]1×l . For this it is necessary to assume a term order on the parameter space which has additional properties (Definition 8). In Section 4.3 we give a division algorithm for F [ N ]1×l . In Section 4.4 we assume that the submonoid N is pointed. The main result for this case, namely Theorem 17, is a stronger version of Theorem 11. 1×θ  = ( θ\ J ), where μ  J = ( = N1× J × N1×(θ\ J ) , μ μϑ )ϑ∈θ = ( μJ,μ μϑ )ϑ∈ J , For J ⊆ θ we identify  N and we write N J = ϑ∈ J Nϑ . 4.1. Compatible data Definition 8. A triple consisting of a finite generating system θ ⊆ Z1×n of N, a subset J ⊆ P (θ) of the power set of θ and a term order  on [l] × N1×θ is compatible if the family (N J ) J ∈J is a conic decomposition of N and the two conditions

∀ J ∈ J ∀(i , μ) ∈ [l] × N J : ρ (i , μ) ∈ [l] × N1× J

(12)

) ∈ [l] × N1×θ ∀( j , ∀ J ∈ J ∀(i , μ ν ) ∈ [l] × N1× J :     )  ( j , )  ρ ψ( j , ν ) =⇒ ρ ψ(i , μ ν) (i , μ

(13)

and

are satisfied. In Section 6, we show that compatible data exist for all finitely generated submonoids N. From now on, we assume that θ , J , and  are compatible. The subset  of [l] × N1×θ carries the induced order. By transferring this order to [l] × N via the bijection (10), we obtain the total order  on [l] × N given by

(i , μ)  ( j , ν ) :⇐⇒ ρ (i , μ)  ρ ( j , ν ) for (i , μ), ( j , ν ) ∈ [l] × N .

ρ mutually inverse order preserving bijections. Therefore, we have that     ∀ p ∈ F [ N ]1×l : deg( p ) = ψ deg χ ( p ) and deg χ ( p ) = ρ deg( p ) ,     ∀ p ∈ F () : deg( p ) = ρ deg ϕ ( p) and deg ϕ ( p ) = ψ deg( p) ,

(14)

This makes ψ| and

(15)

where the degree is the maximal element of the support with respect to  or , depending on the argument. Notice that, since ρ is in general not linear, the resulting order  is not a group order. Lemma 9. Assume compatible data according to Definition 8. Then the total order  we defined in (14) is a generalised term order on [l] × N with respect to the conic decomposition (N J ) J ∈J . Proof. Let (i , μ) ∈ [l] × N. Since  is a term order we have that (i , 0) = ρ (i , 0)  ρ (i , μ) and (14) implies that (i , 0)  (i , μ). To show that (2) is satisfied, let J ∈ J , (i , μ) ∈ [l] × N, ( j , ν ) ∈ [l] × N J and η ∈ N J with (i , μ)  ( j , ν ). Let  η ∈ N1× J be such that ψ( η) = η. From (i , μ)  ( j , ν ) follows that ρ (i , μ)  ρ ( j , ν ).

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Furthermore, since ( j , ν ) ∈ [l] × N J , Equation (12) implies that ρ ( j , ν ) ∈ [l] × N1× J and consequently also ρ ( j , ν ) +  η ∈ [l] × N1× J . Therefore, we can apply Equation (13) and obtain that

 



 







ρ ψ ρ (i , μ) +  η  ρ ψ ρ ( j, ν ) +  η .





But ψ ρ (i , μ) +  η = ψ ρ (i , μ) + ψ( η) = (i , μ) + η = (i , μ + η), and, similarly, ψ ( j , ν + η). Therefore, we conclude using (14) that (i , μ + η)  ( j , ν + η). 2





ρ ( j, ν ) +  η =

4.2. Gröbner bases Assume compatible data and the generalised term order from (14). Let U ⊆ F [ N ]1×l be a submodule. It gives rise to the disjoint decomposition

[l] × N = G  D ,

where D := deg(U ) and G := ([l] × N ) \ deg(U ).

(16)

Let  U := ϕ −1 (U ) ⊆ F [s]1×l . This submodule induces the disjoint decomposition

[l] × N1×θ =  G  D,

where  D := deg( U ) and  G := ([l] × N1×θ ) \ deg( U ).

(17)

The decomposition (17) motivates the following decomposition of [l] × N: Since I 1N×l = ϕ −1 (0) ⊆  U, we have that deg( I 1N×l ) ⊆  D and with Lemma 3 we infer that  G ⊆ ([l] × N1×θ ) \ deg( I 1N×l ) =  . Thus    = G  ( ∩ D ) and, since ψ| is bijective, we obtain the disjoint decomposition

[l] × N = ψ() = ψ( G )  ψ( ∩  D ).

(18)

As we will see in the following lemma, this is actually the same decomposition as (16).





U holds. Furthermore, D = deg(U ) = ψ( ∩  D ) and G = ψ( G ). Lemma 10. The identity U = ϕ F () ∩ 



−1

we infer that χ (U ) ⊆ im(χ ) ∩ Proof. From χ = ϕ | F ()    ϕ χ (U )) ⊆ ϕ F () ∩  U ⊆ ϕ ( U ) = U , i.e., U = ϕ F () ∩  U . From this, we infer that

deg(U ) = deg

 

ϕ F () ∩  U

ϕ −1 (U ) = F () ∩  U . Therefore, U =

(15)   ∗ = ψ deg F () ∩  U = ψ( ∩  D ),

where the equality ∗ follows from the identity deg( F () ∩  U) =  ∩  D, which we prove below. Once this is shown, the disjoint decompositions (16) and (18) imply also the assertion G = ψ( G ). ) ∈ U) ⊆  ∩  D =  ∩ deg( U ) is trivially true. As for the other one, let (i , μ The inclusion deg( F () ∩  ). According to (4), decompose  ∩ D and let  p∈ U with deg( p ) = (i , μ p into  p = p 1 + p 2 ∈ F [s]1×l = F () ⊕ deg( I 1N×l ). From deg( p ) ∈  follows that  p 1 = 0 and deg( p ) = deg( p 1 ). From I 1N×l ⊆  U we infer that  p1 =  p − p2 ∈  U , i.e., we have that  p 1 ∈ F () ∩  U . Therefore, deg( p ) = deg( p 1 ) ∈ deg( F () ∩  U ). 2 Theorem 11. Let Gbe the reduced Gröbner basis of  U with respect to  and define the sets

G := ϕ (G∩ F () ),  G ( J ) := g ∈ G ; deg( g ) ∈ [l] × N J , and  D ( J ) := min p deg(U ) ∩ ([l] × N J ) for all J ∈ J , where the set of minima in the last line is taken with respect to the artinian partial order  p on [l] × N J which is given by

(i , μ)  p ( j , ν ) :⇐⇒ i = j and ν ∈ μ + N J

for (i , μ), ( j , ν ) ∈ [l] × N J .

(19)

Then D ( J ) ⊆ deg(G ( J )), and the equalities

deg(U ) ∩ ([l] × N J ) =

(deg( g ) + N J ) = D ( J ) + N J

g ∈G ( J )

(20)

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hold. As a direct consequence, we obtain that





deg(U ) ∩ ([l] × N J ) = deg(U ) ∩ ([l] × N J ) + N J . Equation (20), together with the fact that (N J ) j ∈J is a conic decomposition of N, implies the disjoint decomposition

[l] × N =



⊆

deg(U )

⊆

⊆

G

[l] × N J = G ∩ ([l] × N J )  D ( J ) + N J . Therefore, in [l] × N J , the subsets deg(U ) ∩ ([l] × N J ) = D ( J ) + N J and G ∩ ([l] × N J ) are the points above and below the staircase, respectively, and the area above the staircase is given by the minimal set D ( J ) in the same fashion as in the classical case N = N1×n . Furthermore, we have that



G=



G ∩ ([l] × N J )

J ∈J

deg(U ) =

(D( J ) + N J ) =

J ∈J

and





deg( g ) + N J .

g ∈G , J ∈J , deg( g )∈[l]×N J

The last identity signifies that G = ϕ (G∩ F () ) is a Gröbner basis of U with respect to the conic decomposition (N J ) J ∈J and the generalised term order . Proof. The main part of the proof consists in showing the identity



deg(U ) ∩ ([l] × N J ) =

deg( g ) + N J



for all J ∈ J .

g ∈G ( J )

⊇. Let g ∈ G ( J ) and η ∈ N J . Thus deg( g ) ∈ [l] × N J , and Lemma 9 and Equation (3) imply that deg( g σ η ) = deg( g ) + η ∈ [l] × N J . Since g σ η ∈ U , we have shown that deg( g ) + η ∈ deg(U ) ∩ ([l] × N J ). −1  ⊆. Let (i , μ) ∈ deg(U ) ∩ ([l] × N J ), and let p ∈ U with deg( p ) = (i , μ). Then χ ( p ) ∈ ϕ (U ) = U . Since G is a Gröbner basis of  U , there are  g ∈ G and  η ∈ N1×θ such that deg χ ( p ) = deg( g) +  η. Since

deg( g) +  η = deg



(15) 

(12)

χ ( p ) = ρ deg( p ) ∈ [l] × N1× J ,

1× J we have that deg( g ) ∈ [l] × N1× J and  . η∈N  g ) ∈ deg( I 1N×l ), then ρ deg( p ) ∈ deg( g ) + N1× J ⊆ deg( I 1N×l ) too, which contradicts If deg(

ρ (i , μ) ⊆ im(ρ ) =  = ([l] × N1×θ ) \ deg( I 1N×l ). Lem. 3

Therefore, deg( g ) ∈  and, altogether, deg( g ) ∈  ∩ ([l] × N1× J ). ) := deg( g ). Since G reduced, we have that Write (i , γ 

 g = sγδi − (sγδi )nf ∈ F () + F (G ) ⊆ F () + F () = F () . Thus, g := ϕ ( g ) ∈ G ( J ) and

deg( p )

(15)

=

(21), (15)

=

   ψ deg( p ) = ψ deg( g) +  η = ψ deg( g ) + ψ( η) deg( g ) + ψ( η) ∈ deg( g ) + N J .

(21)

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To show that D ( J ) ⊆ deg(G ( J )), let (i , μ) ∈ D ( J ). Since D ( J ) ⊆ deg(U ) ∩ ([l] × N J ), there are, by what we have just shown, g ∈ G ( J ) and η ∈ N J with (i , μ) = deg( g ) + η . But this means that (i , μ)  p deg( g ) ∈ deg(U ) ∩ ([l] × N J ). By the definition of D ( J ) as the minimal set of this, we have that (i , μ) = deg( g ).

From this, the inclusion D ( J ) + N J ⊆ deg(G ( J )) + N J = g ∈G ( J ) deg( g ) + N J follows directly. As for the other inclusion, let deg( g ) + η ∈ deg(G ( J )) + N J and let deg( g 1 ) ∈ D ( J ) with deg( g 1 )  p deg( g ). Thus there is an η1 ∈ N J with deg( g ) = deg( g 1 ) + η1 . All in all, deg( g ) + η = deg( g 1 ) + (η + η1 ) ∈ D ( J ) + N J . The remaining assertions follow immediately from what we have shown. 2 Corollary 12. Under the assumptions of Theorem 11, the set



G  := σ μ δi − (σ μ δi )nf ; (i , μ) ∈ J ∈J D ( J ) is the unique reduced Gröbner basis of U with respect to (N J ) J ∈J and .



Proof. Let U  := g ∈G  F [ N ] g ⊆ U be the module generated by the elements of G  . Then deg(U  ) ⊆ deg(U ), and because of

deg(U  ) ⊆ deg(U )

Thm. 11

=

 J ∈J

=





D( J ) + N J =

deg( g σ μ );





deg( g ) + N J



g ∈G  , J ∈J deg( g )∈[l]×N J

μ ∈ N J ⊆ deg(U  ),

g ∈G  , J ∈J deg( g )∈[l]×N J

all those sets are equal. This means that G  is a Gröbner basis of U  and that deg(U  ) = deg(U ). The direct sum decompositions 





and

⊆



=

=

F [ N ]1×l = F ([l]× N )\deg(U ) ⊕ U  F [ N ]1×l = F ([l]× N )\deg(U ) ⊕ U 

imply that U  = U . It remains to prove that G  is reduced. Clearly, all elements of G  have leading coefficient  equal to one. Let g (i ,μ) := σ μ δi − (σ μ δi )nf ∈ G  . We have to show that supp( g (i ,μ) ) = {(i , μ)} ∪ supp (σ μ δi )nf and deg(U (i ,μ) ) are disjoint, where U (i ,μ) :=



g ∈G  \{ g (i ,μ) }

F [ N ] g ⊆ U . The support of the normal form

lies in ([l] × N ) \ deg(U ), which is a subset of ([l] × N ) \ deg(U (i ,μ) ). Now assume that

(i , μ) ∈ deg(U (i ,μ) ) ⊆ deg(U ) =





D( J ) + N J ,

J ∈J

i.e., there are J ∈ J , g ∈ G  with deg( g ) ∈ deg(U ) ∩ ([l] × N J ), and η ∈ N J with (i , μ) = deg( g ) + η . Thus deg( g )  p (i , μ), and since (i , μ) ∈ deg(G  ) ∩ ([l] × N J ) = D ( J ), we conclude that deg( g ) = (i , μ). But this means that g (i ,μ) = g ∈ G  \ { g (i ,μ) }, which is a contradiction. 2 Theorem 11 suggests the following method for computing a Gröbner basis of U with respect to

(N J ) J ∈J and . Algorithm 13. Let U ⊆ F [ N ]1×l be generated by the rows of the matrix R ∈ F [ N ]k×l , i.e., U = F [ N ]1×k R. Compute with Algorithm 6, 4, or with Algorithm 7 a preimage  R ∈ F [s]k×l with ϕ ( R ) = R. Use Algorithm 6, 1, to obtain generators  p1 , . . . ,  p r of the lattice ideal I N ⊆ F [s]. Then

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⎛

⎞

 R ⎜ p 1 idl ⎟

 U = F [s]1×(k+lr ) R,

where  R := ⎝ .. ⎠ ∈ F [s](k+lr )×l .

(22)

.

 pr idl

  Compute the reduced Gröbner basis G of U with respect to . Identify with Algorithm 6 or 7 those    g ∈ G with deg( g ) = ρ ψ deg( g ) . The images ϕ ( g ) of these  g form a Gröbner basis G of U with respect to (N J ) J ∈J and . Proof. Via ϕ , the submodules U = F [ N ]1×k R of F [ N ]1×l are in one-to-one correspondence with those submodules of F [s]1×l which contain ker(ϕ ) = I 1N×l , i.e.,  U = ϕ −1 (U ) = F [s]1×k R + I 1N×l , where  R∈ ϕ −1 ( R ) is an arbitrary preimage of R. Therefore, Equation (22) holds. In the proof of Theorem 11 we have seen that for  g ∈ G the two conditions  g ∈ F () and g ) ∈  are equivalent and the latter deg( condition is satisfied if and only if deg( g ) = ρ ψ deg( g ) . By Theorem 11, the computed set G is indeed a Gröbner basis of U . 2 4.3. A division algorithm We assume compatible data θ , J , and  for N and use the induced generalised term order  from (14) on N with respect to the conic decomposition (N J ) J ∈J . In this section, we show how to divide p ∈ F [ N ]1×l by the Gröbner basis G of U , in particular, how to find the normal form of p.

Lemma 14. Let  p ∈ F [s]1×l and let  p nf be the normal form of  p with respect to  U and . Then ϕ ( p nf ) is the normal form of ϕ ( p ) with respect to U , (N J ) J ∈J , and .  p nf ∈ F (G ) follows that Proof. From 

ϕ ( p nf ) ∈ F

 ψ( G ) Lem. 10

=





F (G ) = F ([l]× N )\deg(U ) .

Furthermore,  p − p nf ∈  U implies that ϕ ( p ) − ϕ ( p nf ) ∈ ϕ ( U ) = U . These two conditions  characterise the normal form with respect to U , (N J ) J ∈J , and  uniquely; therefore ϕ ( p nf ) = ϕ ( p ) nf . 2 Algorithm 15. Let p ∈ F [ N ]1×l . A representation p = zR + p nf , i.e., a tuple z ∈ F [ N ]1×k and the normal form p nf of p with respect to U , (N J ) J ∈J , and  can be computed as follows. Find a preimage

 p ∈ ϕ −1 ( p ) with Algorithm 6 or 7. Use the usual Gröbner bases methods to compute the normal form  p nf with respect to  U and  as well as ( z, z1 , . . . , zr ) ∈ F [s]1×(k+lr ) such that  p = ( z, z1 , . . . , zr ) R + p nf . Then p nf = ϕ ( p nf ) and z = ϕ ( z ). Proof. Application of

ϕ to  p = ( z, z1 , . . . , zr ) R + p nf leads to

⎛R⎞ 0 p = ϕ ( p ) = ϕ ( z, z1 , . . . , zr )ϕ ( R ) + ϕ ( p nf ) = ϕ ( z), ϕ ( z1 ), . . . , ϕ ( zr ) ⎝ .. ⎠ + ϕ ( p nf ) . 

0

z) R + p nf , = ϕ ( where we used Lemma 14 for the last equality.

2

4.4. Pointed submonoids In this section, we will show how Gröbner bases G of U can be computed when the submonoid N is pointed. Of course, Theorem 11 holds also in this case. But while it is essential in the general case that the Gröbner basis G of  U = ϕ −1 (U ) is reduced and that only ϕ ( g ) with  g ∈ G∩ F () comprise G , we will see in Theorem 17 that these requirements are not needed if N is pointed.

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We choose J = {θ}. Therefore, the family (N J ) J ∈J = (Nθ) = ( N ) is trivially a conic decomposition and a relation  is a generalised term order on [l] × N with respect to this decomposition if and only if it is a term order on [l] × N. The compatibility condition (12) does obviously hold and condition (13) simplifies to

    ), ( j , )  ( j , )  ρ ψ( j , ∀(i , μ ν ) ∈ [l] × N1×θ : (i , μ ν ) =⇒ ρ ψ(i , μ ν) .

(23)

From now on, we assume that θ , J = {θ}, and  are compatible. We will show at the end of this section how to obtain suitable term orders . Let  be the induced term order on [l] × N, see (14) and Lemma 9. 7 16. p ∈ F [s]1×l with deg( p ) ∈  the identity Lemma  The map order preserving. As a consequence, for   ψ is ψ deg( p ) = deg ϕ ( p ) holds.

), ( j ,  Proof. Let (i , μ ν ) ∈ [l] × N1×θ . Then

  (14) (23) )  ( j , )  ρ ψ( j , )  ψ( j , (i , μ ν ) =⇒ ρ ψ(i , μ ν ) =⇒ ψ(i , μ ν ), i.e., the map ψ preserves the order.  p ∈ F [s]1×l with deg( p ) ∈  . From deg( p ) ∈  we infer using Lemma 2 that ψ deg( p) ∈ Let     supp ϕ ( p ) . Let ( j , ν ) ∈ supp ϕ ( p ) ⊆ ψ supp( p ) and let ( j ,  ν ) ∈ supp( p ) with ψ( j ,  ν ) = ( j , ν ). Since ψ is order preserving, the fact that ( j ,  ν )  deg( p ) implies that ( j , ν ) = ψ( j ,  ν )  ψ deg( p) . From this, the assertion follows. 2 Theorem 17. Let U ⊆ F [ N ]1×l be a submodule and let  U := ϕ −1 (U ). Let G be an arbitrary Gröbner basis of     U with respect to . Then G := ϕ (G ) is a Gröbner basis of U with respect to , i.e., deg(U ) = deg(G  ) + N, which is what Equation (5) simplifies to in this context. U with respect to . Then, by Theorem 11, G := Proof. Let G be the reduced Gröbner basis of 

ϕ (G∩ F () ) is a Gröbner basis of U , i.e., we have that deg(U ) = deg(G ) + N. χ (g) ∈  U and since deg( U ) = deg(G )+ N1×θ g  ∈ G and  η ∈ N1×θ Let g ∈ , there  are   G . Then    with deg χ ( g ) = deg( g ) + η . Lemma 16 implies that deg ϕ ( g ) = ψ deg( g ) and 

deg( g ) = ψ deg









χ ( g ) = ψ deg( g  ) + ψ( η)

  g  ) + N ⊆ deg ϕ (G ) + N = deg(G  ) + N . ∈ deg ϕ (

Altogether, we have shown that deg(U ) = deg(G ) + N ⊆ deg(G  ) + N ⊆ deg(U ), i.e., all those sets are equal and thus, G  is a Gröbner basis of U . 2 Term orders  which satisfy (23) can be constructed in the following way. Let  be any term order on [l] × N and let 1 be an arbitrary term order on [l] × N1×θ . Then the relation  on [l] × N1×θ defined by

)  ( j , ) < ψ( j , (i , μ ν ) :⇐⇒ ψ(i , μ ν)  ) = ψ( j , ) 1 ( j , or ψ(i , μ ν ) and (i , μ ν) is a term order which satisfies (23).

7

Recall that only the restriction ψ| is order preserving in the general case.

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To obtain a term order  on N, denote the lexicographic term order on R1×h by lex and choose linearly independent maps f 1 , . . . , f h ∈ HomQ (Q N , R), where Q N is the rational vector space spanned by N, such that

f 1 ( N )  0 and

∀i ∈ [h] \ {1} : f i

i −1 



ker( f j ) ∩ N  0.

j =1

If h is large enough, then the relation  on N defined by

μ  ν :⇐⇒ ( f 1 , . . . , f h )(μ) lex ( f 1 , . . . , f h )(ν ), is a term order. In fact, all term orders on N can be found in this way. This result is known as Robbiano’s Theorem (Robbiano, 1985, Thm. 4). In Robbiano (1985), it is formulated for term orders on the positive orthant N1×n ⊆ QN1×n = Q1×n , but the proof holds just as well for arbitrary pointed submonoids N ⊆ Q N. Let F ∈ Rn×h be a matrix with ( f 1 , . . . , f h )(μ) = μ F for all μ ∈ Q N. Then μ  ν if and only if  1  μ F lex ν F . Similarly, let 1 be a term order on N1×θ and let F 1 ∈ Rθ×h1 be such that μ ν ⇐⇒ ,   F 1 lex  μ ν F 1 for μ ν ∈ N1×θ . As in Section 3.2, we interpret θ ∈ Zm×n as a matrix and identify N1×θ = N1×m . Then

θ <  θ F


   θF μ ν ⇐⇒ μ 







F 1 lex  ν θF

F1 ,

i.e., the block matrix θ F F 1 describes the compatible term order  obtained from  and 1 in (24). To obtain term orders on [l] × N from those on N, one proceeds similarly to the polynomial case. For example, if  is a term order on N, then the order given by

 (i , μ)  ( j , ν ) :⇐⇒ i < j or i = j and μ  ν

is a position-over-term order on [l] × N induced by . It should be mentioned that there are term orders on [l] × N which cannot be obtained as position-over-term or term-over-position orders from a term order on N, an example being the orders used in Berkesch and Schreyer (2015, Eq. (1.4) and Cor. 1.11) for the computation of syzygy modules. 5. The Cauchy problem Here we apply Gröbner bases to the solution of the Cauchy problem or initial value problem for systems of inhomogeneous linear partial difference equations over a finitely generated submonoid N. We use all the concepts and the notation we have introduced so far. The monomials σ μ , μ ∈ N, act on the space W := F N  w = w (ν ) ν ∈ N of sequences via left shifts, i.e., (σ μ ◦ w )(ν ) = w (μ + ν ). This action extends to one of F [ N ] on F N in the natural way and makes F N an F [ N ]-module. Thus, a matrix R ∈ F [ N ]k×l induces the map

R ◦ : W l −→ W k , w −→ R ◦ w =

l  j =1

Rij ◦ w j

 i ∈[k]

.

We consider systems of inhomogeneous linear equations R ◦ w = v for given data R ∈ F [ N ]k×l and v ∈ W k and unknown w ∈ W l . Let L ∈ F [ N ]q×k be a universal left annihilator of R, i.e., the rows of L generate the relation module or syzygy module of the rows of R, namely

F [ N ]1×q L = { p ∈ F [ N ]1×k ; p R = 0}.

(25)

The equation R ◦ w = v has a solution if and only if the integrability conditions L ◦ v = 0 are satisfied. It is clear that the condition L ◦ v = 0 is necessary for the existence of a solution. To show that it is

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indeed sufficient one needs the fact that the signal space W = F N an injective module. For a proof that F N is injective, see Oberst (1990, Cor. 12 on p. 52) and use the identification Hom F ( F [ N ], F ) = F N as in Oberst (1990, Ex. 8 on pp. 50–51). For a proof of the fact that over an injective module W , the solvability of R ◦ w = v is equivalent to L ◦ v = 0, see Oberst (1990, Lem. and Def. 31 on pp. 23–24). L of a matrix  R over the polynomial ring F [s] is a The computation of a universal left annihilator  standard task and algorithms for this purpose are included in every computer algebra programme. In the following lemma we present a method for computing universal left annihilators of matrices over the monoid algebra F [ N ]. L ∈ F [s]h×(k+lr ) be a universal left annihilator of the matrix  R from (22), i.e., F [s]1×h L= Lemma 18. Let  1×(k+lr )  h×k   { p ∈ F [s] ; p R = 0}. Let L := L −i i ∈[k] ∈ F [s] be the matrix which consists of the first k columns

of  L. Then ϕ ( L ) ∈ F [ N ]h×k is a universal left annihilator of R.

L = ( L , L 1 , . . . , L r ) as a block matrix Proof. We have to show that Equation (25) holds. We write  where the first block has k columns and all the others have l columns each.

⊆. From  L R = 0 follows that  L R+

r

i =1

 L i idl  p i = 0 and thus



ϕ ( L ) R = ϕ ( L )ϕ ( R ) = ϕ ( L R) = ϕ −

r 

  L i pi .

i =1

Since  p i ∈ I N , the entries of the argument of ϕ in the last expression are obviously in I N = ker(ϕ ) and thus ϕ ( L ) R = 0. ⊇. Let q ∈ F [ N ]1×k with qR = 0. Choose a preimage  q ∈ F [s]1×k with ϕ ( q) = q. From ϕ ( q R) = ϕ ( q)ϕ ( R ) = qR = 0 follows that  q R ∈ ker(ϕ ) = I 1N×l . Since the  p i δ j , i ∈ [r ], j ∈ [l], form a genqi j such that  q R=− erating system of I 1N×l , there are coefficients 



q i j p i δ j . Define the (i , j )∈[r ]×[l]   row  q :=  q , q11 , . . . , q1l , . . . , qr1 , . . . , qrl ∈ F [s]1×(k+lr ) . Then  q R = 0. Since  L is a universal left annihilator of  R there is  z ∈ F [s]1×h such that  q = z L. We extract the first k columns of this expression and obtain  q = z L. Therefore, q = ϕ ( q) = ϕ ( z)ϕ ( L ) ∈ F [ N ]1×h ϕ ( L ). 2

Let  be a generalised term order on [l] × N with respect to a conic decomposition ( N J ) J ∈J .

Consider the module U = F [ N ]1×k R ⊆ F [ N ]1×l generated by the rows of R, its degree set D = deg(U ) and the complement G = ([l] × N ) \ deg(U ). The zero input operator Hs is the matrix

   Hs = Hs (i , μ), ( j , ν )  such that (σ μ δi )nf =



(i ,μ),( j ,ν ) ∈([l]× N )×G

∈ F ([l]× N )×G

 Hs (i , μ), ( j , ν ) σ ν δ j for (i , μ) ∈ [l] × N ,

(26)

( j ,ν )∈G

i.e., its (i , μ)-th row consists of the coefficients of the normal form of given as

(Hs x)i (μ) =



σ μ δi . Its action on x ∈ F G is

 Hs (i , μ), ( j , ν ) x j (ν ) for (i , μ) ∈ [l] × N .

( j ,ν )∈G

To introduce the zero state operator H , let H(i ,μ) ∈ F [ N ]1×k be such that

σ μ δi − (σ μ δi )nf = H(i,μ) R and define

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   H = H (i , μ), (q, η)  

such that H(i ,μ) =

(i ,μ),(q,η) ∈([l]× N )×([k]× N )

∈ F ([l]× N )×([k]× N )

 H (i , μ), (q, η) σ η δq for (i , μ) ∈ [l] × N .

(27)

(q,η)∈[k]× N

The action of this matrix on v ∈ ker( L ◦) ⊆ W k is given by



(H v )i (μ) =

 H (i , μ), (q, η) v q (η) for (i , μ) ∈ [l] × N .

(q,η)∈[k]× N

While the zero input operator Hs is unique, the representation of σ μ δi − (σ μ δi )nf is not unique in general and consequently the matrix H is not unique either. However, the map v −→ H v for v ∈ ker( L ◦) is unique. The two matrices Hs and H have infinitely many rows and columns8 but in each row only finitely many entries are nonzero, i.e., Hs and H are row finite. Theorem 19 (compare Oberst and Pauer, 2001, Thm. 5). Let R ∈ F [ N ]k×l and v ∈ ( F N )k . Assume that the equation R ◦ w = v is solvable, i.e., that L ◦ v = 0, where L is a universal left annihilator of R. Then the following holds: 1. The map





∼ =

w ∈ ( F N )l ; R ◦ w = 0 −→ F G ,

w −→ w |G

is an isomorphism of F -vector spaces. 2. For any initial data x ∈ F G there is a unique solution w of the Cauchy problem R ◦ w = v with w |G = x, namely, w = Hs x + H v. Explicitly, for (i , μ) ∈ [l] × N, this means that



w i (μ) =



 Hs (i , μ), ( j , ν ) x j (ν ) +

 H (i , μ), (q, η) v q (η).

(q,η)∈[k]× N

( j ,ν )∈G

Proof. The proof of this theorem is given in Oberst and Pauer (2001, Thm. 5). We show that the formula w = Hs x + H v is valid for w ∈ ( F N )l with R ◦ w = v, L ◦ v = 0 and w |G = x. This formula implies directly that the solution w for  the given initial values x is unique. Let (i , μ) ∈ [l] × N. From w i (μ) = σ μ δi ◦ w (0) and σ μ δi = (σ μ δi )nf + H(i ,μ) R follows that

w i (μ)

= = (26), (27)

=

 (σ μ δi )nf ◦ w (0) + (H(i ,μ) R ) ◦ w (0)   μ (σ δi )nf ◦ w (0) + H(i ,μ) ◦ ( R ◦ w ) (0)    



 Hs (i , μ), ( j , ν ) σ ν δ j ◦ w (0)

( j ,ν )∈G

+ =





  H (i , μ), (q, η) σ η δq ◦ v (0)

(q,η)∈[k]× N

( j ,ν )∈G

=

=v



 Hs (i , μ), ( j , ν ) w j (ν ) +   

(Hs x)i (μ) + (H v )i (μ).

= x j (ν )



 H (i , μ), (q, η) v q (η)

(q,η)∈[k]× N

2

As the theorem demonstrates, the complement G of the degree set of deg(U ) is the initial value region, i.e., the values of the solution can be prescribed freely on G and determine the solution uniquely.

8

In special cases, the matrix Hs has only finitely many columns.

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For N = N and if N is interpreted as (discrete) time axis one expects the initial values to be given on discrete time intervals starting at zero, and this condition is satisfied by the sets G for N = N. Formulated differently, if (i , μ) ∈ G and (i , μ) = (i , ν ) + η , then (i , ν ) ∈ G too, in other words, the set G is closed under taking predecessors. For arbitrary submonoids N, a generalisation of this condition holds, namely, for all J ∈ J , the sets G ∩ ([l] × N J ) are closed under taking predecessors with respect to the partial order  p on [l] × N J from (19). However, in contrast to the case N = N, it may happen that the initial value area G is interspersed with points of deg(U ). For example, if N = N2 + N3 = {0, 2, 3, 4, 5, 6, . . . } and deg(U ) = 4 + N = {4, 6, 7, 8, 9, . . . }, then the initial value area is G = {0, 2, 3, 5}, where 1 is missing because it is not in the monoid N and 4 is missing because 4 ∈ deg(U ), which means that the solution at the point 4 is determined by the values of the solution at the points 0, 2, 3, and 5. 6. The construction of generalised term orders and compatible data For arbitrary finitely generated submonoids N ⊆ Z1×n we construct a conic decomposition and a generalised term order on N with respect to it. Then we use this to derive compatible data for N. With a generalised term order on N, one can easily form a generalised position-over-term or termover-position term order on [l] × N and the corresponding compatible data. We denote the vector space, (rational) convex cone, abelian group, and submonoid generated by a set X ⊆ Q1×n by Q X , Q0 X , Z X , and N X , respectively. Furthermore, the dual space of a rational vector space V is denoted by V ∗ = HomQ ( V , Q). 6.1. The construction of a generalised term order Since we consider N instead of [l] × N here, the conditions that a total order  is a generalised term order with respect to a conic decomposition ( N J ) J ∈J simplify to 0 = min N and

 ∀ J ∈ J ∀μ ∈ N ∀ν , η ∈ N J : μ  ν =⇒ μ + η  ν + η .

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 Let θ  ⊆ Z1×n be finite with N = of N based on the Nθ . We will derive a conic decomposition  facets of the polytope conv θ ∪ {0} , where conv denotes the convex hull in Q1×n . We recall several facts on polyhedra, see, e.g., Schrijver (1986, pp. 101–102). A subset F of a polyhedron P ⊆ Q1×n is a face if and only if there are g ∈ (Q P )∗ and c ∈ Q such that g ( P )  c and g ( F ) = c and then we have that F = P ∩ g −1 (c ). In particular, this implies that the faces are polyhedra. A face F induces the finitely generated convex cone

 N F := g ∈ (Q P )∗ ; g ( F ) = max{ g (ν ); ν ∈ P } in the dual space (Q P )∗ , i.e., N F is the dual cone of the convex cone spanned by F . A facet is a face of affine dimension dim( F ) = dim( P ) − 1. This means that dim(N F ) = 1, i.e., N F = Q0 g for some g ∈ (Q P )∗ \ {0}. For all facets F , choose g F ∈ N F \ {0}. Then P can be written as

P=

 



μ ∈ Q P ; g F (μ)  max{ g F (ν ); ν ∈ P } .

F facet

Assume that 0 ∈ P . Then we have for all g ∈ (Q P )∗ and c ∈ Q with g ( P )  c that c  0. Furthermore, for a facet F of P with 0 ∈ F , all g ∈ N F satisfy that max{ g (ν ); ν ∈ P } = 0. On the other hand, 0 ∈ /F is equivalent to the (unique) existence of a map f F ∈ N F with max{ f F (ν ); ν ∈ P } = 1. We define the two sets

F := { F ⊆ P facet; 0 ∈ / F}

and

F0 := { F ⊆ P facet; 0 ∈ F }.

For F ∈ F0 choose an arbitrary map f F ∈ N F \ {0}. Then P can be written as

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P=

{μ ∈ Q P ; f F (μ)  1} ∩

F ∈F



{μ ∈ Q P ; f F (μ)  0}

(29)

F ∈F0

and all F ∈ F satisfy that F = P ∩ f F−1 (1).  From now on, let P be the polytope P = conv θ  ∪ {0} . The convex cone generated by P is Q0 P = Q0 θ  = Q0 N. We define the map

f : Q0 P −→ Q,

μ −→ max{ f F (μ); F ∈ F }.

This map has the following basic properties. Lemma 20. 1. Let μ ∈ Q0 P and a ∈ Q0 . Then f (aμ) = af (μ). 2. For μ, ν ∈ Q0 P we have that f (μ + ν )  f (μ) + f (ν ). Proof. 1. Let F , F  ∈ F with f (μ) = f F (μ) and f (aμ) = f F  (aμ). The assertion follows from

f (aμ) = f F  (aμ) = af F  (μ)  af (μ) = af F (μ) = f F (aμ)  f (aμ). 2. Let F ∈ F with f (μ + ν ) = f F (μ + ν ). Then

f (μ + ν ) = f F (μ + ν ) = f F (μ) + f F (ν )  f (μ) + f (ν ).

2

The following lemma is a preparation for Lemma 22, where a decisive property of f is shown. Lemma 21. Let X ⊆ Q1×n be a convex set. Then Q0 X =



η∈ X Q0 η .

 ∈ Q(X0) be a coefficient μ ∈ Q0 X and let μ   ϑ ϑ . Set a := ϑ∈ X μ ϑ . Then η := 1a μ = ϑ∈ X μaϑ ϑ is a convex combination vector with μ = ϑ∈ X μ of elements in X and thus η ∈ X . Furthermore, μ = aη ∈ Q0 η . 2 Proof. The inclusion ⊇ is clear. As for the other one, let





Lemma 22. Let μ ∈ P with f (μ)  0. Then μ = 0. From this follows immediately that μ = 0 is equivalent to f (μ) > 0 for μ ∈ P . This implies that f ( P )  0 and, with Lemmas 21 and 20, 1, that f (Q0 P )  0. Proof. The assumption f (μ)  0 means that f F (μ)  0 for all F ∈ F . This implies that f F (Q0 μ)  0 and, in particular, that f F (Q0 μ)  1 for all F ∈ F . In addition, from μ ∈ P we infer for all F ∈ F0 that f F (μ)  0 and consequently f F (Q0 μ)  0. Therefore, Equation (29) is satisfied for all elements of Q0 μ, i.e., Q0 μ ⊆ P . Since P is a polytope, it is bounded. If μ = 0 then Q0 μ is a ray and thus an unbounded subset of a bounded set, which is a contradiction. Therefore, we conclude that μ = 0. 2 For a facet F ∈ F we consider the convex cone C F := Q0 F . To decide if a cone C F we can use the map f , as we show in the following lemma.

μ ∈ Q0 P lies in a

Lemma 23. Let F ∈ F . Then C F = {μ ∈ Q0 P ; f F (μ) = f (μ)}. Proof.

⊆. Let μ ∈ C F . Since the facet F is convex, Lemma 21 implies that there are ν ∈ F and a ∈ Q0 such that μ = aν . Since ν ∈ P we have that f F  (ν )  1 for all F  ∈ F . Furthermore, ν ∈ F implies that f F (ν ) = 1. Together this means that f (ν ) = f F (ν ) and, using Lemma 20, 1, we have that f (μ) = f (aν ) = af (ν ) = af F (ν ) = f F (aν ) = f F (μ).

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⊇. Let μ be in the set on the right hand side. In the case μ = 0 the assertion follows trivially. If μ = 0 then we infer from Lemma 22 that f (μ) > 0 and we consider ν := f (1μ) μ. Using the assumption that f F (μ) = f (μ) we have that fF (ν ) = 1. We will show that ν ∈ P . On one hand, for all F  ∈ F we have that f F  (ν )  f (ν ) = f f (1μ) μ = f (1μ) f (μ) = 1. On the other hand, since P is convex, Lemma 21 implies that μ ∈ Q0 P can be written as μ = aη with η ∈ P and a ∈ Q0 . Since η ∈ P we have for all F  ∈ F0 that f F  (η)  0 and, since ν = f (aμ) η and f (aμ)  0, we infer that f F  (ν )  0. Equation (29) implies that and consequently μ = f (μ)ν ∈ Q0 F = C F .

ν ∈ P . Altogether, we have that ν ∈ P ∩ f F−1 (1) = F 2

Corollary 24.



1. Q0 P = F ∈F C F . 2. The restrictions f |C F are linear maps for all F ∈ F . 3. Let μ, ν ∈ Q0 P with f (μ + ν ) = f (μ) + f (ν ). By item 1, there exists an F ∈ F with μ + ν ∈ C F . Then both μ and ν are in C F . Proof. 1. The inclusion ⊇ is clear. As for the other one, let μ ∈ Q0 P and let F ∈ F with f (μ) = f F (μ). Lemma 23 implies that μ ∈ C F . 2. Let μ, ν ∈ C F . Since C F is a convex cone, μ + ν lies in C F too. From Lemma 23 follows that f (μ) = f F (μ), f (ν ) = f F (ν ) and f (μ + ν ) = f F (μ + ν ). Since f F is linear, the assertion follows. 3. We have that f (μ) + f (ν ) = f (μ + ν ) = f F (μ + ν ) = f F (μ) + f F (ν ). Since f F (μ)  f (μ) and f F (ν )  f (ν ), this implies that f F (μ) = f (μ) and f F (ν ) = f (ν ) and from Lemma 23 follows that μ ∈ C F and ν ∈ C F . 2 Lemma 25. 1. Let C ⊆ Q1×n be a finitely generated convex cone. Then C ∩ Z1×n is a finitely generated submonoid of Z1×n . This result is known as Gordan’s Lemma, see, for example, Cox et al. (2011, Prop. 1.2.17) or Miller and Sturmfels (2005, Thm. 7.16). 2. Let N 1 , N 2 ⊆ Z1×n be finitely generated submonoids. Then N 1 ∩ N 2 is also a finitely generated submonoid. Proof. For a proof of Gordan’s Lemma we refer to the literature. To show the second assertion, let N i = Nθi for finite θi ⊆ Z1×n and i = 1, 2. The set



1×(θ1 θ2 )

 := ( 2 ) ∈ Q0 M μ1 , μ

;



1ϑ ϑ = μ

ϑ∈θ1



2ϑ ϑ μ



ϑ∈θ2

is the solution set of finitely many homogeneous linear inequalities over Q1×(θ1 θ2 ) and therefore a  ∩ Z1×(θ1 θ2 ) ⊆ N1×(θ1 θ2 ) finitely generated convex cone. By Gordan’s Lemma, the submonoid M := M is finitely generated. Let θˆ ⊆ N1×(θ1 θ2 ) be finite with M = Nθˆ . The map

ξ : Z1×(θ1 θ2 ) −→ Z1×n ,

2 ) −→ ( μ1 , μ



1ϑ ϑ, μ

ϑ∈θ1

is linear and satisfies that

    1ϑ ϑ = 2ϑ ϑ = N 1 ∩ N 2 . 2 ) ∈ N1×(θ1 θ2 ) : μ = ξ( M ) = μ ∈ Z1×n ; ∃( μ1 , μ μ μ ϑ∈θ1

ϑ∈θ2

Furthermore, the image of a submonoid under a linear map is generated by the images of the generators, i.e., N 1 ∩ N 2 = ξ( M ) = ξ(Nθˆ ) = Nξ(θˆ ), which proves the assertion. 2

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M. Scheicher / Journal of Symbolic Computation ••• (••••) •••–•••

22

We consider the submonoids N F = C F ∩ N = Q0 F ∩ N ⊆ Z1×n , where F ∈ F . Lemma 26. 1. Let M ⊆ Z1×n be a finitely generated submonoid. Then Q0 M =





η∈M . 2. Let F ∈ F . Then C F =



1 η∈ N F Q0 η = a η; a ∈ N \ {0},

η∈ M Q0 η =

1 a

η; a ∈ N \ {0},



η ∈ NF .

Proof. (M )

 ∈ Q0 with μ = 1. Both inclusions ⊇ are trivial. To show the equality, let μ ∈ Q0 M and let μ    ϑ ∈ N, therefore μ ϑ . Let a ∈ N \ { 0 } be a common denominator of the μ , ϑ ∈ M. Then aμ ϑ ϑ ϑ∈ M  ϑ ϑ ∈ M and μ = 1a η ∈ Q0 η . η := aμ = ϑ∈M aμ 2. Again, the inclusions ⊇ are clear. Let μ ∈ C F and, by item 1, let a ∈ N \ {0} and η ∈ N with μ = 1a η. Since μ ∈ C F and C F is a cone we have that η ∈ C F , i.e., η ∈ C F ∩ N = N F . 2 Let Z N be a total group order on Z N. On N we define the relation  by





μ  ν :⇐⇒ f (μ) < f (ν ) or f (μ) = f (ν ) and μ ZN ν .

(30)

Theorem 27. The family ( N F ) F ∈F is a conic decomposition of N and  is a generalised term order on N with respect to ( N F ) F ∈F . Proof. We start by showing that ( N F ) F ∈F is a conic decomposition.



F ∈F

N F = N. Corollary 24, 1, implies that

NF =

F ∈F



F ∈F

C F = Q0 P = Q0 N. Therefore,

C F ∩ N = Q0 N ∩ N = N .

F ∈F

N F is finitely generated. Since C F is a finitely generated convex cone, Gordan’s Lemma implies that C F ∩ Z1×n is a finitely generated submonoid. The submonoid N is also finitely generated. By Lemma 25, 2, the intersection C F ∩ Z1×n ∩ N = C F ∩ N = N F is finitely generated, too. N F is pointed. Let μ ∈ U( N F ). Then μ ∈ N F ⊆ C F and Lemma 23 implies that f (μ) = f F (μ). Likewise, from −μ ∈ N F follows that f (−μ) = f F (−μ) = − f F (μ) = − f (μ). Since μ, −μ ∈ N F ⊆ Q0 P , Lemma 22 implies that f (μ), f (−μ)  0. Therefore, f (μ) = f (−μ) = 0 and, again with Lemma 22, we conclude that μ = 0. Z N F = Z N. The inclusion ⊆ is clear. Before we show the other one, notice for F ∈ F that 0 is in C F but not in the affine hull of F (which is equal to f F−1 (1)). Together with dim( F ) = dim(Q0 N ) − 1 this implies that dim(C F ) = dim(Q0 N ) = dim( N ), i.e., QC F = Q N. Since C F is a convex cone, we have that Q N = QC F = C F − C F . Let μ ∈ Z N = N − N and write μ = μ1 − μ2 with μi ∈ N. Since μi ∈ N ⊆ Q N = C F − C F we decompose further and write μi = νi − ηi with νi , ηi ∈ C F . Now we use Lemma 26, 2, to obtain representations νi = a1 ζi and ηi = b1 ξi with ai , b i ∈ N \ {0} and ζi , ξi ∈ N F . Therefore, i

i

μ1 + ξ1 + ξ2 = a11 ζ1 + b1b−1 1 ξ1 + ξ2 ∈ C F ∩ N = N F . 



∈N







∈C F



μ2 + ξ1 + ξ2 ∈ N F . Thus we conclude that  μ = μ1 − μ2 = μ1 + ξ1 + ξ2 − μ2 + ξ1 + ξ2 ∈ N F − N F = Z N F .

Similarly, we obtain that



Now we show that  is a generalised term order. The relation is clearly a partial order and it is total because Z N is total.

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M. Scheicher / Journal of Symbolic Computation ••• (••••) •••–•••

23

min N = 0. Let μ ∈ N. From Lemma 22 we infer that f (μ)  0. In the case that f (μ) > 0 we have that f (0) = 0 < f (μ) and this means that 0 < μ. If f (μ) = 0 then Lemma 22 implies that μ = 0 and therefore 0  0 = μ.  satisfies (28). Let F ∈ F , μ ∈ N, and ν , η ∈ N F with μ  ν . In the case that f (μ) < f (ν ) we have that

f (μ + η)

Lem. 20, 2



f (μ) + f (η) < f (ν ) + f (η)

Cor. 24, 2

=

f (ν + η)

and this implies that μ + η  ν + η . In the case that f (μ) = f (ν ) the same computation we have just performed furnishes f (μ + η)  f (ν + η). Also, we have that μ Z N ν and, since Z N is a group order, μ + η Z N ν + η . Put together, this means that μ + η  ν + η . 2 6.2. The construction of compatible data Now we will construct compatible data for the submonoid N = Nθ  ⊆ Z1×n , i.e., a suitable finite generating system θ ⊆ Z1×n , a subset J ⊆ P (θ) of the power set of θ such that the family (N J ) J ∈J is a conic decomposition of N and a term order  on N1×θ such that the two conditions

∀ J ∈ J ∀μ ∈ N J : ρ (μ) ∈ N1× J and       ∀ J ∈ J ∀ μ ∈ N1×θ ∀ ν ∈ N1× J : μ ν =⇒ ρ ψ( μ)  ρ ψ( ν)

(31) (32)

hold, where ρ is the section of ψ defined as in (9). These conditions are just the conditions (12) and (13) for N instead of [l] × N. We start with a finite generating system θ  of N and use all the data we derived in the previous section. For F ∈ F let θ F be a finite generating system of the submonoid N F . Set θ := F ∈F θ F \ {0} and let ψ be the parametrisation associated with θ as in (6). For F ∈ F define J F := θ ∩ N F and J := { J F ; F ∈ F }. Clearly, N J F = N F . In fact, the map F → J , F → J F , is a bijection, but we do not need this here. We define the linear map

  = ( f : N1×θ −→ Q0 , μ μϑ )ϑ∈θ −→



ϑ f (ϑ). μ

ϑ∈θ

Because of f ( N )  0, we have that  f (N1×θ )  0. The map  f has the following properties. Lemma 28.





 ∈ N1×θ the relation  f ( μ)  f ψ( μ) holds. 1. For all μ

2. For F ∈ F , the restriction of  f to N1× J F is  f |N1× J F = ( f ◦ ψ)|N1× J F .  1 ×θ ∈N  ∈ N 1× J F . 3. Let μ with  f ( μ) = f ψ( μ) and let F ∈ F with ψ( μ) ∈ N F . Then μ Proof. 1. The assertion follows from

 f ( μ) =



ϑ f (ϑ) μ

Lem. 20, 1

=

ϑ∈θ



f ( μϑ ϑ)

ϑ∈θ

1× J F

Lem. 20, 2



f











ϑ ϑ = f ψ( μ μ) .

ϑ∈θ

∈N  = ( ϑ ϑ . We have that ϑ ∈ J F ⊆ N F ⊆ C F 2. Let μ , i.e., μ μϑ )ϑ∈ J F , and thus ψ( μ) = ϑ∈ J F μ and by Corollary 24, 2, the map f is linear on C F . Therefore,





f ψ( μ) = f



ϑ∈ J F



ϑ ϑ = μ

 ϑ∈ J F

ϑ f (ϑ) =  μ f ( μ).

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24

3. From the assumption and from Lemma 20, 1, we infer that

f











ϑ ϑ = f ψ( μ μ) = f ( μ) =

ϑ∈θ



ϑ f (ϑ) = μ

ϑ∈θ



f ( μϑ ϑ)

ϑ∈θ

ϑ ϑ = ψ( ϑ ϑ ∈ C F for all ϑ ∈ θ . and μ) ∈ N F ⊆ C F . Thus Corollary 24, 3, implies that μ ϑ∈θ μ ϑ = 0 then ϑ ∈ C F and, since ϑ ∈ θ ⊆ N, this means that ϑ ∈ C F ∩ N ∩ θ = N F ∩ θ = Therefore, if μ ϑ = 0 for all ϑ ∈ θ \ J F , in other words, μ  ∈ N 1× J F . 2 J F . This implies that μ Let 1 be an arbitrary term order on N1×θ . We define the relation  on N1×θ via

  μ ν :⇐⇒ f ( μ) < f ( ν)





or  f ( μ) = f ( ν ) and ψ( μ)
   1  or  f ( μ) = f ( ν ) and ψ( μ) = ψ( ν ) and μ ν .

(33)

Theorem 29. The triple θ , J , and  is compatible. Proof. Since N J F = N F and the family ( N F ) F ∈F is a conic decomposition of N (see Theorem 27), the set J has all the necessary properties. Furthermore,  is clearly a total group order. It remains to show that min N1×θ = 0 and that  satisfies the conditions (31) and (32).

 ∈ N1×θ . If    0. Now assume that  min  N1×θ = 0. Let μ f ( μ) > 0 = f (0) then μ f ( μ) =  μ f (ϑ) = 0. All the summands are products of nonnegative numbers, therefore for all ϑ∈θ ϑ ϑ = 0 or f (ϑ) = 0. If μ ϑ = 0 then f (ϑ) = 0 and, by Lemma 22, this means ϑ ∈ θ we have that μ ϑ = 0 cannot happen and consethat ϑ = 0. But 0 ∈ / θ by the way we defined θ . Therefore, μ  = 0 and, in particular, 0  0 = μ . quently, we have that μ  satisfies (31). Let J F ∈ J and μ ∈ N J F . We will show that ρ (μ) ∈ N1× J F by proving that ρ (μ) ∈ /  ∈ ψ −1 (μ) ∩ N1× J F = ∅. From N1×θ \ N1× J F . To do this, let  ν ∈ ψ −1 (μ ) with  νθ\ J F = 0 and let μ  Lemma 28, 2, it follows that  f ( μ) = f ψ( μ) = f (μ) = f ψ( ν ) . Since  ν ∈/ N1× J F , items 1 and 



 ≺ 3 of Lemma 28 imply that f ψ( ν ) < f ( ν ). Therefore f ( μ) < f ( ν ), and this implies that μ ν. − 1 Thus  ν = min ψ (μ) = ρ (μ).   ∈ N1×θ and     satisfies (32). Let J F ∈ J , μ ν ∈ N1× J F with μ ν . We have to show that ρ ψ( μ)   1 × J ρ ψ( ν ) . From  ν ∈ N F follows that ψ( ν ) ∈ N J F and, by the previous part of this proof, ρ ψ( ν ) ∈ N1× J F . Thus

   Lem. 28, 2   Lem. 28, 2  f ρ ψ( ν) = f ψ ρ (ψ( ν )) = f ψ( ν) =  f ( ν ).    =ψ( ν)

     implies that  Furthermore, ρ ψ( μ)  μ f ρ ψ( μ)  f ( μ). Using this and μ ν , we obtain 

that

     f ρ ψ( μ)  f ( μ)  f ( ν ) = f ρ ψ( ν) .

(34)     If  f ρ ψ( μ) < f ρ ψ( ν ) then the assertion follows. On the other hand, if the two expres   sions are equal, then  f ( μ) = f ( ν ) and, since μ ν , this implies that μ) Z N ψ( ν ). Since ψ ◦ ρ  ψ( is the identity function on N, this means that ψ ρ ψ( μ ) Z N ψ ρ ψ( ν ) .     In the sub-case ψ ρ ψ( μ)
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25

7. Examples In this section we give comprehensive examples to the concepts and algorithms of this article. In Example 30 we demonstrate the method of Section 6 on how to construct a generalised term order and compatible data for a general submonoid N. Then we compute a Gröbner basis of a submodule of F [ N ]1×2 and solve the Cauchy problem. In Sections 7.1 and 7.2 we consider the case that N is the whole integer grid, i.e., that N = Z1×n . In Section 7.1 we show that if one uses the decomposition of Z1×n into orthants—which is a conic decomposition of Z1×n —, then it is very easy to find compatible data. Example 30. Let N ⊆ Z1×2 be the submonoid generated by the set

 θ  := (1, 0), (2, 1), (0, 2), (0, −2), (4, 0) ⊆ Z1×2 , see Fig. 1a for a sketch. The last (4, 0) is superfluous in θ  for the generation of N, but it  generator makes the polytope P = conv θ  ∪ {0} symmetric with respect to the x-axis and this simplifies the expressions in this example considerably. The polytope P has the three facets F 1 , F 2 , F 3 which are plotted in Fig. 1a and we group them into the sets F = { F 1 , F 2 } and F0 = { F 3 }, where F consists of those facets which do not contain zero and F0 is made up of those who contain zero. For the facets F in F , the maps f F with f F ( F ) = 1 and f F ( P )  1 are

f F 1 , f F 2 : Q N = Q1×2 −→ Q,

f F 1 (μ) = μ



1/4 1/2





,

1/4



f F 2 (μ) = μ −1/2 .

The facets in F give rise to the pointed convex cones





C F 1 = Q0 (1, 0), (0, 1)





C F 2 = Q0 (1, 0), (0, −1) ,

and

which are the grey shaded areas in Fig. 1a, and those, in turn, are used to obtain the submonoids





N F 1 = C F 1 ∩ N = Nθ F 1 with θ F 1 = (1, 0), (0, 2), (2, 1)



and



N F 2 = C F 2 ∩ N = Nθ F 2 with θ F 2 = (1, 0), (0, −2), (2, −1) , which comprise a conic decomposition of N. The linear maps f F 1 and f F 2 induce the map

f : Q0 N −→ Q,





 

μ −→ max f F 1 (μ), f F 2 (μ) = max μ

1/4 1/2



,μ



1/4 −1/2

 ,

1×2 which we use, after choosing a total group order Z1×2 on Z , for defining the generalised term order  on N with respect to N F 1 , N F 2 as in (30). The union θ = θ F 1 ∪ θ F 2 is a generating system of N and the intersections J F i = θ ∩ N F i generate N F i for i = 1, 2, see Fig. 1b. In this example the identity J F i = θ F i holds, but only the inclusion ⊇ can be guaranteed in general. The values of the elements of θ under f are

ϑ ∈θ f (ϑ)

(1, 0) 1/4

(0, 2) 1

(0, −2)

(2, 1)

(2, −1)

1

1

1

and this gives rise to the linear map

  −→ f : N1×θ −→ Q, μ



ϑ f (ϑ) = 14 μ (1,0) + μ (0,2) + μ (0,−2) + μ (2,1) + μ (2,−1) . μ

ϑ∈θ

Once a group order Z1×2 on Z1×2 and a term order 1 on N1×θ are chosen, this map induces the term order  as in (33) and the data θ , J F 1 , J F 2 and  are compatible.

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M. Scheicher / Journal of Symbolic Computation ••• (••••) •••–•••

26

Fig. 1. The submonoid N from Example 30. The elements of the submonoid are marked by • and the generators are additionally marked by ◦. The pointed convex cones C F 1 and C F 2 are shaded in light and dark grey, respectively.

To make this computationally more accessible, we interpret θ as a matrix in Z5×2 and identify

N1×θ = N1×5 , i.e.,

⎛1

θ=

⎜0 ⎝0 2 2

⎛

⎞

0 2 ⎟ −2 ⎠ 1 −1

and thus  f (μ) = μ Xf with Xf =

⎞

1/4 ⎜ 1 ⎟ ⎝ 1 ⎠. 1 1

Let Z1×2 be the graded reverse lexicographic order on Z1×2 , i.e.,

μ Z1×2 ν ⇐⇒ μ X Z1×2 lex ν X Z1×2 with X Z1×2 =



1 0 1 −1

 ,

and let 1 be the lexicographic term order on N1×5 , which is represented by the identity matrix id5 . Then the term order  is given by

1 /4  ⎜ 1 ⎜ id5 = ⎜ 1 ⎝ 1 1







⎛

   X lex  μ ν ⇐⇒ μ ν X with X = Xf θ X Z1×2

1 0 2 −2 −2 2 3 −1 1 1

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

⎞

0 0⎟ ⎟ 0 ⎟. ⎠ 0 1 (35)

The last three columns of X are linear combinations of the first five and thus can be omitted. Let ψ and ϕ be the parametrisations induced by θ as in (6) and (7), respectively. By Algorithm 6, 1, the lattice ideal of N with respect to θ is

IN =

4 

F [s] p i with  p 1 = s2 s5 − s4 ,  p 2 = s3 s4 − s5 ,  p 3 = s 2 s 3 − 1,  p 4 = s41 − s4 s5 .

i =1

We use the term order  from (35) to define a position-over-term order on [2] × N1×5 which we denote by the same symbol, namely,

   )  ( j , (i , μ ν ) :⇐⇒ i < j or i = j and μ ν .

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27

Let U = F [ N ]1×3 R ⊆ F [ N ]1×2 be the submodule generated by the rows of the matrix

⎞

⎛

σ22 +σ1 0 2 2 −1 ⎝ σ σ σ14 +σ12 σ2 +σ2−2 ⎠ ∈ F [ N ]3×2 . R= 1 2 +σ 1 σ 2 σ 3 σ2 +σ13 σ2−1 +σ22 +σ1 σ15 +σ13 σ2 +σ1 σ2−2

The preimage of this module under

ϕ is  U = ϕ −1 (U ) = F [s]1×11 R with

⎛  ⎞ R ⎜ ⎟ ⎜ p 1 id2 ⎟ 11×2 ⎟  R =⎜ , where  R= ⎜ .. ⎟ ∈ F [ N ] ⎝ . ⎠

s2 +s1

0

s4 +s5

s41 +s4 +s3

! ∈ F [s]3×2

s1 s4 +s1 s5 +s2 +s1 s51 +s1 s4 +s1 s3

 p 4 id2

is a preimage of R. The reduced Gröbner basis of  U with respect to  is the set

 G= (s2 + s1 , 0), (s1 s3 + 1, 0), (s1 s5 + s4 , 0), (s25 + s31 , 0), (s24 + s51 , 0), (s4 + s5 , s41 + s4 + s3 ), (s3 s5 + s5 , s25 + s23 + s5 ), (s4 + s5 , s4 s5 + s4 + s3 ), (−s1 s4 + s4 , s24 + s2 s4 + 1), (s3 s4 − s5 , 0), (s4 s5 − s41 , 0), (0, s2 s3 − 1), (0, s3 s4 − s5 ), (0, s2 s5 − s4 ) consisting of 14 elements. The last five elements lie in I 1N×2 and the support of the sixth element is not contained in F () . Therefore and according to Theorem 11, we have that

G = ϕ (G∩ F () )  = (σ22 + σ1 , 0), (σ1 σ2−2 + 1, 0), (σ13 σ2−1 + σ12 σ2 , 0), (σ14 σ2−2 + σ13 , 0),

(σ14 σ22 + σ15 , 0), (σ12 σ2−3 + σ12 σ2−1 , σ14 σ2−2 + σ2−4 + σ12 σ2−1 ),

(σ12 σ2 + σ12 σ2−1 , σ14 + σ12 σ2 + σ2−2 ), (−σ13 σ2 + σ12 σ2 , σ14 σ22 + σ12 σ23 + 1)

is a Gröbner basis of U with respect to ( N F 1 , N F 2 ) and the generalised position-over-term order on [2] × N induced by the generalised term order  on N from (30). From this, we can read off the degree set of U , namely

deg(U ) =

 ∪ ∪





1, (0, 2) + N F 1 ∪









1, (4, −2) + N F 2 ∪







2, (4, 0) + N ∪





1, (1, −2) + N F 2 ∪











1, (4, 2) + N F 1 ∪







1, (3, −1) + N F 2







2, (4, −2) + N F 2



2, (4, 2) + N F 1 .

The degree of the sixth element of G lies in {2} × N F 1 as well as in {2} × N F 2 and because of this, both submonoids have to be added to it. The degree set of U and the degrees of the elements of G are sketched in Fig. 2. To divide the monomial σ17 σ2−1 δ2 ∈ F [ N ]1×2 with respect to U , ( N F 1 , N F 2 ), and , the first step is to compute a preimage, namely, s1 s4 s25 δ2 ∈ ϕ −1 (σ17 σ2−1 δ2 ). We divide this preimage with respect to  U and , i.e., we compute the representation

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28

Fig. 2. The degree set of the module U from Example 30. The first component is plotted in the left graph, the second in the right one. The points in the degree set are marked by •, the ones not in the degree set by ×. In addition, the degrees of the Gröbner basis elements are marked by ◦.

 z1

 z2

0

 z3











 =  z

0 0

0



s1 − 1 −s1 s5 + s1  R







s5 − 1 0





 = −s25 + s5 s1 s5 − s1 0 



s1 s4 s25 δ2 = ( z,  z1 ,  z2 ,  z3 ,  z4 ) R + (s1 s4 s25 δ2 )nf

 z4

+ −s51 + s1 s4 + s41 − s4 −s1 s3 s5 + s1 s4 + s1 s3 .    

=(s1 s4 s25 δ2 )nf

Now we apply the map

ϕ to this and obtain





σ17 σ2−1 δ2 = ϕ ( z,  z1  z2 ,  z3 ,  z4 )ϕ ( R ) + ϕ (s1 s4 s25 δ2 )nf = ϕ ( z) R + (σ17 σ2−1 δ2 )nf  = −σ14 σ2−2 + σ12 σ2−1 σ13 σ2−1 − σ1 0 R    

=ϕ ( z)=: z

3 −3 1 2

+ −σ + σ σ + σ − σ σ −σ σ   5 1

3 1 2

4 1

2 1 2

=(σ17 σ2−1 δ2 )nf

−2

+ σ σ + σ1 σ2 3 1 2







(36)

,

i.e., a representation σ17 σ2−1 δ2 = zR + (σ17 σ2−1 δ2 )nf . The identity ϕ (s1 s4 s25 δ2 )nf = (σ17 σ2−1 δ2 )nf holds because of Lemma 14. Finally, we consider the Cauchy problem for the system of equations R ◦ w = v with unknown w, inhomogeneity v and initial conditions w |G = x with the initial value region G = ([2] × N ) \ deg(U ). The tuple v ∈ ( F N )2 has to satisfy the integrability conditions L ◦ v = 0, where L is a universal left annihilator of R. To obtain such a matrix L, one has to compute a universal left annihilator  L of  R.

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29

The first three columns of  L correspond to the rows of  R, and consequently, also to the rows of R. According to Lemma 18, the images of its these columns under ϕ form a universal left annihilator L of R. In L, rows equal to zero can be omitted. Since  L includes also the relations between the generators of I N as well as those between them and the rows of  R , it is rather unwieldy and therefore we do not print it. However, the resulting universal left annihilator



L= 1



σ1 −1 ∈ F [ N ]1×3

of R is simple.   Assume that initial values x j (ν ) ( j ,ν )∈G as well as an inhomogeneity v = v q (η) (q,η)∈[3]× N with

L ◦ v = 0 are given. We want to compute the value of the solution w at the position (i , μ) = 2, (7, −1) , i.e., w 2 (7, −1). From Equation (36) we can read off the relevant entries of the zero input

and the zero state operator, namely

( j , ν ) 1, (5, 0) 1, (3, 1) 1, (4, 0) 1, (2, 1) 2, (3, −3 ) 2, (3, 1) 2, (1, −2) else

 Hs 2, (7, −1) , ( j , ν ) −1 1 1

−1 −1

(q, η) 1, (4, −2) 1, (2, −1) 2, (3, −1 ) 2, (1, 0) else

 H 2, (7, −1) , (q, η) −1 1 1

−1 0

1 1 0

From this, we conclude that

w 2 (7, −1) = −x1 (5, 0) + x1 (3, 1) + x1 (4, 0) − x1 (2, 1) − x2 (3, −3) + x2 (3, 1)

+ x2 (1, −2) − v 1 (4, −2) + v 1 (2, −1) + v 2 (3, −1) − v 2 (1, 0). 7.1. Subgroups: The decomposition into orthants Here and in Section 7.2 we consider the case that N = U( N ) is a subgroup of Z1×n . Since Z1×n is a free abelian group, the subgroup N is also free and thus isomorphic to Z1×k for some k  n, see Lang (2002, p. 38 and Thm. 7.3). For this reason, it is no loss of generality to investigate only the case that N = Z1×n is the whole integer grid. We consider the generating system θ = {δ j ; j ∈ [n]} ∪ {−δ j ; j ∈ [n]} of N = Z1×n . It gives rise to the parametrisation

ψ : N1×θ −→ Z1×n ,



 −→ μ δ j − μ −δ j μ

j ∈[n]

.

Let J := { J ⊆ θ; ∀ j ∈ [n] : either δ j ∈ J or − δ j ∈ J }. The N J , J ∈ J , are exactly the orthants of Z1×n . ± For μ ∈ Z1×n let μ± ∈ N1×θ be defined by μ± δ := max{0, μ j } and μ−δ := − min{0, μ j } for j ∈ [n]. The image of

μ± under ψ is ψ(μ± ) = μ.

j

Lemma 31. 1. Let μ ∈ N = Z1×n . Then ψ −1 (μ) ⊆ μ± + N1×θ . 2. Let J ∈ J and μ ∈ N J . Then ψ −1 (μ) ∩ N1× J = {μ± }.

j

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Proof.

 ∈ N1×θ , the 1. In Zerz and Oberst (1993, Eq. (13) on p. 256), it has been shown that for all μ  = ψ( −δ j } for j ∈ [n]. In identity μ μ)± + γ holds, where γ is defined by γδ j := γ−δ j := min{ μδ j , μ

 ∈ ψ( particular, we have that μ μ)± + N1×θ . Writing μ = ψ( μ) ∈ Z1×n , we obtain that ψ −1 (μ) ⊆ μ± + N1×θ . 2. First, we show that μ± ∈ N1× J . Let j ∈ [n] with δ j ∈ θ \ J . Then −δ j ∈ J and μ j  0. Consequently, μ± δ = max{0, μ j } = 0. Similarly, for j ∈ [n] with −δ j ∈ θ \ J we have that δ j ∈ J and μ j  0 and j

± μ± −δ j = − min{0, μ j } = 0. All in all, we have that μϑ = 0 for all ϑ ∈ θ \ J and this ± 1× J means that μ ∈ N .  ∈ ψ −1 (μ) ∩ N1× J . From item 1 and the previous part of this proof follows that Let μ  = μ± +  ψ −1 (μ) ∩ N1× J ⊆ μ± + N1× J , i.e., we can write μ η for some  η ∈ N1× J . From

consequently

μ = ψ( μ) = ψ(μ± ) + ψ( η) = μ + ψ( η) we infer that ψ( η) = 0, i.e., ψ( η) j =  ηδ j −  η−δ j = 0 for all j ∈ [n]. This, together with the fact that  η ∈ N1× J and consequently  ηδ j = 0 or  η−δ j = 0, means that both  ηδ j and  η−δ j are zero.  = μ± . 2 Summarising, we conclude that  η = 0, i.e., μ Proposition 32. The triple θ , J , and  is compatible for any term order  on [l] × N1×θ . Proof. We have to prove that the conditions (12) and (13) are satisfied. Notice that from Lemma 31, 1, follows that ρ (i , μ) = (i , μ± ) for (i , μ) ∈ [l] × N. (12) is satisfied. Let J ∈ J and (i , μ) ∈ [l] × N J . From Lemma 31, 2, we infer that μ± ∈ N1× J and thus ρ (i , μ) = (i , μ± ) ∈ [l] × N1× J . ) ∈ [l] × N1×θ , and ( j ,  )  ( j ,  ν ) ∈ [l] × N1× J with (i , μ ν ). From  ν∈ (13) is satisfied. Let J ∈ J , (i , μ 1× J ± N we infer using Lemma 31, 2, that  ν = ψ( ν ) and this implies that ( j ,  ν ) = ( j , ψ( ν )± ) = ρ ψ( j ,  ν ) . We conclude that









)  (i , μ )  ( j , ρ ψ(i , μ ν ) = ρ ψ( j , ν) .

2

In Oberst and Pauer (2001, Eq. (97) and Thm. 43 on p. 303), it has been shown that given any term order  on [l] ×N1×θ , the total order  induced via (14) is a generalised term order on [l] ×Z1×n with respect to the conic decomposition (N J ) J ∈J . Here, this follows from Proposition 32 and Lemma 9. Example 33. Let N = Z1×2 and let U = F [ N ] R ⊆ F [ N ] be the ideal generated by the Laurent polynomial R = σ12 σ23 + σ1 σ2 + σ22 − σ1 − 2σ2 . We use the parametrisation and decomposition of N into orthants as described in this section and interpret θ =



id2 − id2



as a matrix in Z4×2 . The lattice ideal

of N with respect to this parametrisation is

I N = F [s](s1 s3 − 1) + F [s](s2 s4 − 1), where F [s] = F [s1 , s2 , s3 , s4 ]. The reduced Gröbner basis of

 U = ϕ −1 (U ) = F [s](s21 s32 + s1 s2 + s22 − s1 − 2s2 ) + I N with respect to the lexicographic term order lex on F [s] is

G=

1

(2s23 s24 − s23 s4 + s3 s34 − s3 s24 − 1), s2 − 2s23 s4 + s23 − s3 s24 + s3 s4 , s1 − 2s3 s24 + s3 s4 − s34 + s24 . 2

All of its elements lie in F () , thus

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  G = ϕ G∩ F () = ϕ (G) = 12 (2σ1−2 σ2−2 − σ1−2 σ2−1 + σ1−1 σ2−3 − σ1−1 σ2−2 − 1),

σ2 − 2σ1−2 σ2−1 + σ1−2 − σ1−1 σ2−2 + σ1−1 σ2−1 , σ1 − 2σ1−1 σ2−2 + σ1−1 σ2−1 − σ2−3 + σ2−2



is a Gröbner basis of U with respect to the conic decomposition of Z1×2 into the quadrants and the generalised term order given by

μ  ν ⇐⇒ ρ (μ) lex ρ (ν ) ⇐⇒ μ± lex ν ± .





From the degree set of G , namely from (−2, −2), (0, 1), (1, 0) , we can determine the degree set of U via Theorem 11. The point (−2, −2) lies in the third quadrant of the plane, (0, 1) lies in the first as well as in the second one and (1, 0) lies in the first and fourth quadrant. Therefore, we obtain







deg(U ) = (−2, −2) + (−N) × (−N)

      ∪ ( 0 , 1 ) + Z × N ∪ (1 , 0 ) + N × Z .

The sets D = deg(U ) and its complement G = Z1×2 \ deg(U ) are plotted in Fig. 3a. As always with Gröbner bases, choosing a different term order often leads to a different degree set deg(U ). The reduced Gröbner basis of  U with respect to the graded reverse lexicographic term order on F [s] is

 G= s2 s4 − 1, s1 s3 − 1, s1 s2 − 2s3 s4 − s24 + s3 + s4 , 1 (4s23 s4 − s34 − 2s23 − s3 s4 + s24 4 s1 s34 − s1 s24 − s21 + 2s24 − s4 .

+ s1 − 2s2 ),

1 (2s3 s24 + s34 − s3 s4 − s24 2 s2 s23 + s22 − 2s23 − s3 s4 + s3 ,

− s1 ),

The first two elements of G are elements of the lattice ideal, therefore they are not in F () and do not contribute to the Gröbner basis

  G = ϕ G∩ F () = σ1 σ2 − 2σ1−1 σ2−1 − σ2−2 + σ1−1 + σ2−1 , 1 (2 1−1 2−2 + 2−3 − 1−1 2−1 − 2−2 − 1 ), 2 1 (4 1−2 2−1 − 2−3 − 2 1−2 − 1−1 2−1 + 2−2 4 −2 −2 −1 −1 −1 2 2 1 + 2 −2 1 − 1 2 + 1 , −3 −2 −2 −1 2 1 2 − 1 2 − 1 +2 2 − 2

σ

σ

σ

σ

σ

σ

σ σ

σ

σ σ

σ σ

σ

σ

σ

σ

σ

σ

σ

σ

σ

σ

σ

σ

+ σ1 − 2σ2 ),

σ

σ

σ

of U . The degree set of U is











deg(U ) = (1, 1) + N1×2 ∪ (−1, −2) + (−N)1×2 ∪ (−2, −1) + (−N)1×2



      ∪ (−2, 1) + (−N) × N ∪ (1, −3) + N × (−N) .

The corresponding decomposition of N into the regions above and below the staircase are plotted in Fig. 3b. 7.2. Subgroups: Another classical decomposition We consider the generating system θ  = {δ1 , . . . , δn , δn+1 }, where δ1 , . . . , δn is the standard basis of  Z1×n and δn+1 := − i ∈[n] δi . Then P = conv θ  ∪ {0} is the polytope with vertices δ1 , . . . , δn+1 and  faces F k := conv {δi ; i ∈ [n + 1] \ {k}} for k ∈ [n + 1]. None of the facets contain zero, thus all of them comprise the set F . Consequently, C F k = Q0 {δi ; i ∈ [n + 1] \{k}} and N F k = N{δi ; i ∈ [n + 1] \{k}} and  δi we have that θ = θ  . Since f (μ) = 1 for all vertices μ of P , the map  f is given by  f ( μ) = i∈[n+1] μ  ∈ N1×θ . for μ

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Fig. 3. The degree sets of the module U from Examples 33, 34 and 35 with respect to various parametrisations of the integer lattice Z1×2 and term orders in the parameter space. The points in the degree set are marked by •, the ones not in the degree set by ×. In addition, the degrees of the Gröbner basis elements are marked by ◦.

Now we interpret the elements of θ as the rows of the matrix θ =



idn



−1 ··· −1

. Let Z1×n be a

group order on Z1×n , described by the matrix X Z1×n . This matrix has n rows and is of rank n. Then, ,  for μ ν ∈ N1×θ = N1×(n+1) , we have that

⎛

1

⎞

⎛

⎜.    X lex  μ ν ⇐⇒ μ ν X with X = ⎜ .

⎝1

. ⎟ ⎜ ⎜ .. θ X Z1×n ⎟ ⎟=⎜ ⎠ ⎜ ⎝1

1

1

⎜.

⎞

1 X Z1×n



⎟ ⎟ ⎟. ⎟ ⎠

i ∈[n] ( X Z1×n )i −

1×n The matrix X has rank n + 1 and therefore we do not need the  additional term order 1 on Z to break ties since there are none. By Theorem 29, the data θ , {δi ; i ∈ [n + 1] \ {k}}; k ∈ [n + 1] , and  are compatible.

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Example 34. As in Example 33 let N = Z1×2 and let U = F [ N ] R ⊆ F [ N ] with R = σ12 σ23 + σ1 σ2 +



σ22 − σ1 − 2σ2 . The parametrisation of N is given by the matrix θ =

1 0 0 1 −1 −1



. The lattice ideal is

I N = F [s](s1 s2 s3 − 1), where F [s] = F [s1 , s2 , s3 ]. The conic decomposition of N which we just derived is the one into the three sets

N F 3 = N1×2 ,

N F 2 = N(1, 0) + N(−1, −1),

and N F 1 = N(0, 1) + N(−1, −1).

On Z1×2 , we choose the graded reverse lexicographic order. Then the term order  on N1×3 is de⎛ ⎞ 1 1 0 ⎜ 1 −1 ⎟. scribed by the matrix X = ⎝ 1 ⎠ 1 2 −1 The preimage of U with respect to this parametrisation is

 U = ϕ −1 (U ) = F [s](s21 s32 + s1 s2 + s22 − s1 − 2s2 ) + F [s](s1 s2 s3 + 1). The reduced Gröbner basis of  U with respect to  is

 G= s1 s2 s3 − 1, s22 s3 + s1 s22 − s1 s3 − 2s2 s3 + 1, s21 s22 − s21 s3 + s1 + s2 − 2, s21 s23 − s1 s3 − s2 s3 − s1 s2 + 2s3 . All but the first element are in F () . Therefore,

 G = σ1−1 σ2 + σ1 σ22 − σ2−1 − 2σ1−1 + 1, σ12 σ22 − σ1 σ2−1 + σ1 + σ2 − 2, σ2−2 − σ2−1 − σ1−1 − σ1 σ2 + 2σ1−1 σ2−1 is a Gröbner basis of U with respect to the compatible data. The degree set of U is









deg(U ) = (−1, 1) + N F 1 ∪ (2, 2) + N F 3 ) ∪ (0, −2) + N F 2



and it is plotted in Fig. 3c. The method we described here is not the only way to construct compatible data. For example, the generating system θ and the conic decomposition as before as well as an elimination term order  for sn+1 on N1×θ form compatible data, compare Pauer and Unterkircher (1999, Ex. 2.5). Example 35. We use the data from Example 34 along with the lexicographic term order on F [s] with s3 > s1 > s2 , which is an elimination term order for s3 . The reduced Gröbner basis of  U with respect to this term order is

 G= s21 s32 + s1 s2 − s1 + s22 − 2s2 , s3 s32 − 2s3 s22 + s1 s32 + s2 − 1, s3 s1 − s3 s22 + 2s3 s2 − s1 s22 − 1 and its image under

ϕ forms the Gröbner basis

 G = σ12 σ23 + σ1 σ2 − σ1 + σ22 − 2σ2 , σ1−1 σ22 − 2σ1−1 σ2 + σ1 σ23 + σ2 − 1, σ2−1 − σ1−1 σ2 + 2σ1−1 − σ1 σ22 − 1 of U . The degree set of U is











deg(U ) = (2, 3) + N F 3 ∪ (−1, 2) + N F 1 ∪ (0, −1) + N F 2 and it is plotted in Fig. 3d.



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8. Concluding remarks I have shown that conic decompositions and generalised term orders, as well as compatible data exist for all finitely generated submonoids of the integer lattice Z1×n . Furthermore, I have presented algorithms for the computation of Gröbner bases, of normal forms, and of syzygy modules which use for the main computational task the corresponding well-known algorithms over the polynomial ring. I have implemented these algorithms in the computer algebra system Singular9 and used this for the examples in Section 7. I plan to release them as a library for Singular, but there is still much work to be done. My motivation for this article comes from systems theory, in particular, from the Cauchy problem. Gröbner bases methods may be valuable for the derivation of generating functions of the solutions of the Cauchy problem (Bousquet-Mélou and Petkovšek, 2000; Nekrasova, 2015). In systems theory, there are many other problems which can be solved using Gröbner bases—see, e.g., Scheicher (2016) and the references given therein—, and I hope that many fields of applications will emerge. Some open problems in the further development of the theory and its algorithms are the elimination of variables, the computation of radicals and primary decompositions, and also the more general case that F is only a ring instead of a field. Acknowledgements During the nDS 2013 conference in Erlangen, Anton Kummert and Ulrich Oberst discussed possible initial value regions for a difference equation similar to the one of Examples 33, 34, and 35. Oberst solved the problem using the parametrisation from Section 7.1 and approached me to implement the algorithm and to compute an example. The problem evolved and the result is the present article. I am grateful to Anton Kummert for posing the problem and for debating it with Oberst and myself. I thank Ulrich Oberst for forwarding the problem and him as well as Ingrid Blumthaler for numerous discussions and persistent constructive criticism. Furthermore, I thank the two reviewers for their thorough work. Their criticism improved this article substantially. References Berkesch, C., Schreyer, F.O., 2015. Syzygies, finite length modules, and random curves. In: Eisenbud, D., Iyengar, S.B., Singh, A.K., Stafford, J.T., van den Bergh, M. (Eds.), Commutative Algebra and Noncommutative Algebraic Geometry. Volume I: Expository Articles. Cambridge University Press, Cambridge, pp. 25–52. Bourlès, H., Marinescu, B., Oberst, U., 2015. The injectivity of the canonical signal module for multidimensional linear systems of difference equations with variable coefficients. Multidimens. Syst. Signal Process. http://dx.doi.org/10.1007/ s11045-015-0331-x. Bousquet-Mélou, M., Petkovšek, M., 2000. Linear recurrences with constant coefficients: the multivariate case. Discrete Math. 225 (1–3), 51–75. Conti, P., Traverso, C., 1991. Buchberger algorithm and integer programming. In: Applied Algebra, Algebraic Algorithms and Error-correcting Codes. New Orleans, LA, 1991. In: Lect. Notes Comput. Sci., vol. 539. Springer, Berlin, pp. 130–139. Cox, D., Little, J., Schenck, H., 2011. Toric Varieties. American Mathematical Society, Providence. Kreuzer, M., Robbiano, L., 2000. Computational Commutative Algebra. 1. Springer, Berlin. Lang, S., 2002. Algebra. Springer, New York. Lim, J.S., 1988. Two-dimensional signal processing. In: Lim, J.S., Oppenheim, A.V. (Eds.), Advanced Topics in Signal Processing. Prentice-Hall, Englewood Cliffs, pp. 338–415. Miller, E., Sturmfels, B., 2005. Combinatorial Commutative Algebra. Grad. Texts Math., vol. 227. Springer, New York. Nekrasova, T.I., 2015. On the Cauchy problem for multidimensional difference equations in rational cone. J. Sib. Fed. Univ. Math. Phys. 8 (2), 184–191. Oberst, U., 1990. Multidimensional constant linear systems. Acta Appl. Math. 20, 1–175. Oberst, U., Pauer, F., 2001. The constructive solution of linear systems of partial difference and differential equations with constant coefficients. Multidimens. Syst. Signal Process. 12 (3–4), 253–308. Park, H., Regensburger, G. (Eds.), 2007. Gröbner Bases in Control Theory and Signal Processing. Radon Series on Computational and Applied Mathematics, vol. 3. de Gruyter.

9

http://singular.uni-kl.de.

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Pauer, F., Unterkircher, A., 1999. Gröbner bases for ideals in Laurent polynomial rings and their application to systems of difference equations. Appl. Algebra Eng. Commun. Comput. 9 (4), 271–291. Pauer, F., Zampieri, S., 1996. Gröbner bases with respect to generalized term orders and their application to the modelling problem. J. Symb. Comput. 21 (2), 155–168. Robbiano, L., 1985. Term orderings on the polynomial ring. In: EUROCAL ’85, vol. 2. Linz, 1985. In: Lect. Notes Comput. Sci., vol. 204. Springer, Berlin, pp. 513–517. Scheicher, M., 2016. Gröbner bases over finitely generated affine monoids and applications. The direct sum case. In: Proceedings of the 22nd International Symposium on the Mathematical Theory of Networks and Systems (MTNS). Minneapolis, USA, pp. 174–181. Schrijver, A., 1986. Theory of Linear and Integer Programming. John Wiley & Sons, Chichester. Shannon, D., Sweedler, M., 1988. Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence. J. Symb. Comput. 6 (2–3), 267–273. Sturmfels, B., 1996. Gröbner Bases and Convex Polytopes. American Mathematical Society, Providence. Theis, C., 1999. Der Buchberger-Algorithmus für torische Ideale und seine Anwendung in der ganzzahligen Optimierung. Master’s thesis. Universität des Saarlandes. Zampieri, S., 1994. A solution of the Cauchy problem for multidimensional discrete linear shift-invariant systems. Linear Algebra Appl. 202, 143–162. Zerz, E., Oberst, U., 1993. The canonical Cauchy problem for linear systems of partial difference equations with constant coefficients over the complete r-dimensional integral lattice Zr . Acta Appl. Math. 31 (3), 249–273.