Gr{ö}bner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems
aa r X i v : . [ c s . S C ] F e b Gröbner Basis over Semigroup Algebras: Algorithms andApplications for Sparse Polynomial Systems
Matías R. Bender Jean-Charles Faugère Elias Tsigaridas
Sorbonne Université, CNRS, INRIA, Laboratoire d’Informatique de Paris 6, LIP6, Équipe P olSys
F-75005, Paris, [email protected]
ABSTRACT
Gröbner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problemwith a complexity at least single exponential in the number ofvariables. However, in most of the cases, the polynomial systemscoming from applications have some kind of structure. For exam-ple, several problems in computer-aided design, robotics, vision, bi-ology, kinematics, cryptography, and optimization involve sparsesystems where the input polynomials have a few non-zero terms.Our approach to exploit sparsity is to embed the systems in asemigroup algebra and to compute Gröbner bases over this algebra.Up to now, the algorithms that follow this approach benefit fromthe sparsity only in the case where all the polynomials have thesame sparsity structure, that is the same Newton polytope. We in-troduce the first algorithm that overcomes this restriction. Underregularity assumptions, it performs no redundant computations.Further, we extend this algorithm to compute Gröbner basis in thestandard algebra and solve sparse polynomials systems over thetorus ( C ∗ ) n . The complexity of the algorithm depends on the New-ton polytopes. The introduction of the first algorithm to compute Gröbner basesin 1965 [4] established them as a central tool in nonlinear alge-bra. Their applications span most of the spectrum of mathemat-ics and engineering [5]. Computing Gröbner bases is an intrinsi-cally hard problem. For many “interesting” cases related to appli-cations the complexity of the algorithms to compute them is sin-gle exponential in the number of variables, but there are instanceswhere the complexity is double exponential; it is an
EXPSPACE complete problem [33]. There are many practically efficient algo-rithms, see [11, 16] and references therein, for which, under gener-icity assumptions, we can deduce precise complexity estimates [1].However, the polynomial systems coming from applications, i.e.computer-aided design, robotics, biology, cryptography, and opti-mization e.g., [14, 19, 42], have some kind of structure. One of themain challenges in Gröbner basis theory is to improve the com-plexity and the practical performance of the related algorithms byexploiting the structure.We employ the structure related to the sparsity of the polyno-mial systems; in other words, we focus on the non-zero terms of theinput polynomials. In addition, we consider polynomials havingdifferent supports. There are different approaches to benefit fromsparsity, e.g., [2, 8, 18, 20, 39]. We follow [20, 39] and we considerGröbner bases over semigroup algebras. We construct a semigroupalgebra related to the Newton polytopes of the input polynomials and compute Gröbner bases for the ideal generated by the originalpolynomials in this semigroup algebra.We embed the system in semigroup algebras because in thisplace they “behave” in a predictable way that we can exploit algo-rithmically. Semigroup algebras are related to toric varieties. Anaffine toric variety is the spectrum of a semigroup algebra [10,Thm. 1.1.17]. Hence, the variety defined by the polynomials overthe semigroup is a subvariety of a toric variety. This variety is dif-ferent from the one defined by the polynomials over the originalpolynomial algebra, but they are related and in many applicationsthe difference is irrelevant, e.g., [14]. We refer to [10] for an intro-duction to toric varieties and to [41] for their relation with Gröbnerbasis.In ISSAC’14, Faugère et al. [20] considered sparse unmixed sys-tems , that is, polynomial systems where all the polynomials havethe same Newton polytope, and they introduced an algorithm tocompute Gröbner bases over the semigroup algebra generated bythe Newton polytope. This algorithm is a variant of the MatrixF5 al-gorithm [1, 16]. They compute Gröbner basis by performing Gauss-ian elimination on various Macaulay matrices [30] and they avoidcomputations with rows reducing to zero using the F5 criterion[16]. The efficiency of this approach relies on an incremental con-struction which, under regularity assumptions, skips all the rowsreducing to zero. They exploit the property that, for normal New-ton polytopes, generic unmixed systems are regular sequences overthe corresponding semigroup algebra. Unfortunately, this propertyis no longer true for mixed systems , that is, for systems of polynomi-als with different Newton polytopes. So, this algorithm fails to pre-dict all rows reducing to zero during Gaussian elimination. More-over, the degree bound for the maximal degree in [20, Lem. 5.2]misses some assumptions to hold, see App. A. We relax the reg-ularity assumptions of [20] and we introduce an F5-like criterionthat, under regularity assumptions, predicts all the rows reducingto zero during Gröbner bases computation.In this context, we also mention our previous work [2] on com-puting sparse Gröbner bases for mixed sparse polynomial systems.We emphasize that besides the similarity in the titles, this work and[2] are completely different approaches. First, we compute differ-ent objects. Sparse Gröbner bases [2, Sec. 3] are not Gröbner basisfor semigroup algebras. Moreover, we follow different computa-tional strategies: in [2] we perform the computations polynomialby polynomial, while in this work we proceed degree by degree.Further, when we use [2] to solve 0-dimensional systems, there areno complexity bounds, let alone bounds depending on the Newtonpolytopes, for this approach.A direct application of Gröbner basis theory is to solve poly-nomial systems. This is also an intrinsically hard problem [26]. atías R. Bender, Jean-Charles Faugère, and Elias Tsigaridas
Hence, it is important to exploit the sparsity of the input polyno-mials to obtain new algorithms for solving with better complexitybounds. The different ways of doing so include homotopy meth-ods e.g., [28, 44], chordal elimination [8], triangular decomposition[36], and various other techniques [21, 25, 27, 35, 37, 43].Among the symbolic approaches related to toric geometry, themain tool to solve sparse systems is the sparse resultant [23]. Theresultant is a central object in elimination theory and there aremany different ways of exploiting it to solve sparse systems, seefor example [9, Chp. 7.6]. Canny and Emiris [6] and Sturmfels [40]showed how to compute the sparse resultant as the determinantof a square Macaulay matrix (Sylvester-type formula) whose rowsare related to mixed subdivisions of some polytopes. Using this ma-trix, e.g., [13, 15], we can solve square sparse systems. For this, weadd one more polynomial to the system and we consider the ma-trix of the resultant of the new system. Under genericity assump-tions, we can recover the multiplication maps of the quotient ringdefined by original square system over the ring of Laurent polyno-mials and we obtain the solutions over ( C \ { }) n . Recently, Massri[32] dropped the genericity assumptions by considering a biggermatrix.We build on Massri’s work and, under regularity assumptions,we propose an algorithm to solve 0-dimensional square systemswith complexity related to the Minkowski sum of the Newton poly-topes. Because we work with toric varieties, we compute solutionsover ( C \ { }) n . Our strategy is to reuse part of our algorithm tocompute Gröbner bases over semigroup algebras to compute mul-tiplication maps and, via FGLM [17], recover a Gröbner basis overthe standard polynomial algebra. As we compute the solutions over ( C \ { }) n , we do not recover a Gröbner basis for the original ideal,but for its saturation with respect to the product of all the vari-ables. We compute with a matrix that has the same size as the onein Emiris’ resultant approach [13]. Our approach to solve is moregeneral than the one in [20] as we compute with mixed sparse sys-tems, and because it terminates earlier as we do not compute Gröb-ner bases but multiplication maps. An overview of our strategy isas follows:(1) Let f , . . . , f n ∈ K [ x ] be a sparse regular polynomial systemwith a finite number of solutions over ( C ∗ ) n .(2) Embed the polynomials to a multigraded semigroup algebra K [ S h ∆ ] related to the Newton polytopes of f , . . . , f n and to thestandard n-simplex (see Def. 2.6).(3) For each variable x i : • Use the Gröbner basis algorithm (Alg. 2) to construct a squareMacaulay matrix related to ( f , . . . , f n , x i ) of size equal to thenumber of integer points in the Minkowski sum of the Newtonpolytopes of f , . . . , f n and the n-simplex. • Split the matrix in four parts and compute a Schur complement,which is the multiplication map of x i in K [ x ± ]/h f , . . . , f n i .(4) Use the multiplication maps and FGLM to get a Gröbner ba-sis for h f , . . . , f n i : h Î i x i i ∞ with respect to any monomialorder.The contributions and consequences of our work include: • We introduce the first effective algorithm to compute Gröbner basesover semigroup algebras associated to mixed polynomial systems.
We generalize the work of [20] to the mixed case from which we could provide accurate complexity estimates related to theNewton polytopes of the input polynomials. • We relate the solving techniques using Sylvester-type formulas inresultant theory with Gröbner bases computations.
The simplest,but not necessarily the most efficient as there are more compactformulas [46], way to compute the resultant is to use a Sylvester-type formula and compute it as the determinant of a Macaulaymatrix [9, Chp. 3.4]. Using this matrix we extract multiplicationmaps and solve polynomial systems [9, Chp. 3.4]. In the stan-dard polynomial algebra, such matrices are at the heart of linearalgebra algorithms to compute Gröbner bases because they cor-respond to the biggest matrix that appears during Gröbner basiscomputations for regular 0-dimensional systems [30]. However,such a relation was not known for the sparse case. We bring outthis relation and we build on it algorithmically. • We generalize the F5 criterion to depend on Koszul complexes in-stead of regular sequences.
The exactness of the Koszul complexis closely related to regular sequences [12, Ch. 17] and, geomet-rically, to complete intersections. Roughly speaking, when weconsider generic square systems of equations in the coordinatering of a “nice” projective variety, the variety that the systemdefines is closely related to a complete intersection. In this case,the Koszul complex of the system might not be exact in general,but only in some “low” degrees. Hence, even if the system isnot a regular sequence, by focusing on the degrees at which thestrands of the Koszul complex are exact, we can still predict thealgebraic structure of the system and perform efficient computa-tions. Using this property we extend the classical F5-like criteriathat apply only to regular sequences. Moreover, additional in-formation on the exactness of the strands of the Koszul complexand the multigraded Castelnuovo-Mumford regularity [3, 31] re-sults in better degree and complexity bounds; similarly to thecase of the multihomogeneous systems [2, Sec. 4]. • We disrupt the classical strategy to solve 0-dimensional systemsusing Gröbner basis, by avoiding intermediate Gröbner basis com-putations.
The classical approach for solving 0-dimensional sys-tems using Gröbner bases involves the computation of a interme-diate Gröbner basis that we use to deduce multiplication mapsand, by using FGLM, to obtain the lexicographical Gröbner ba-sis of the ideal. If the intermediate Gröbner basis is computedwith respect to a graded reverse lexicographical order and the in-put system “behaves well” when we homogenize it, this strategyis some sense optimal because it is related to the Castelnuovo-Mumford regularity of the homogenized ideal [7, Cor. 3].However, over semigroup algebras, it might not be always pos-sible to relate the complexity of the intermediate Gröbner ba-sis computation to the Castelnuovo-Mumford regularity of theideal; this is so because we can not define monomial orders thatbehave like a graded reverse lexicographical, see [2, Ex. 2.3]. Weovercome this obstacle by truncating the computation of the in-termediate Gröbner basis in such a way that the complexity isgiven by Castelnuovo-Mumford regularity of the ideal.
Let K ⊂ C be a field of characteristic , x : = ( x , . . . , x n ) , and K [ x ] : = K [ x , . . . , x n ] . We consider : = ( , . . . , ) and : = ( , . . . , ) .For each r ∈ N , let e , . . . , e r be the canonical basis of R r . Given röbner Basis over Semigroup Algebras d , d ∈ N r , we say d ≥ d when d − d ∈ N r . We use [ r ] = { , . . . , r } . We denote by h f , . . . , f m i the ideal generated by f , . . . , f m . Definition 2.1 (Affine semigroup and semigroup algebra).
Follow-ing [34], an affine semigroup S is a finitely-generated additive sub-semigroup of Z n , for some n ∈ N , such that it contains ∈ Z n .An affine semigroup S is pointed if it does not contain non-zeroinvertible elements, that is for all α , β ∈ S \ { } , α + β , [34,Def 7.8]. The semigroup algebra K [ S ] is the K -algebra generatedby the monomials { X α : α ∈ S } such that X α · X β = X α + β . Definition 2.2 (Convex set and convex hull).
A set ∆ ⊂ R n isconvex if every line segment connecting two elements of ∆ alsolies in ∆ ; that is, for every α , β ∈ ∆ and ≤ λ ≤ it holds λ α + ( − λ ) β ∈ ∆ . The convex hull of ∆ is the unique minimal, withrespect to inclusion, convex set that contains ∆ . Definition 2.3 (Pointed rational polyhedral cones).
A cone C is aconvex subset of R n such that ∈ C and for every α ∈ C and λ > , λ α ∈ C . The dimension of a cone is the dimension of thevector space spanned by the cone. A cone is pointed if does notcontain any line; that is, if , α ∈ C , then − α < C . A ray is apointed cone of dimension one. A ray is rational if it contains anon-zero point of Z n . A rational polyhedral cone is the convex hullof a finite set of rational rays. For a set of points ∆ ⊂ R n , let C ∆ bethe cone generated by the elements in ∆ . If ∆ is (the convex hullof) a finite set of integer points, then C ∆ is a rational polyhedralcone.A rational polyhedral cone C defines the affine semigroup C ∩ Z n , which is pointed if and only if the cone is pointed. Definition 2.4 (Integer polytopes and Minkowski sum).
A integerpolytope ∆ ⊂ R n is the convex hull of a finite set of (integer) pointsin Z n . The Minkowski sum of two integer polytopes ∆ and ∆ is ∆ + ∆ = { α + β : α ∈ ∆ , β ∈ ∆ } . For each polytope ∆ and k ∈ N , we denote by k · ∆ the Minkowski sum of k copies of ∆ . Definition 2.5 (Laurent polynomials and Newton polytopes).
ALaurent polynomial is a finite K -linear combination of monomials X α , where α ∈ Z n . The Laurent polynomials form a ring, K [ Z n ] ,that corresponds to the semigroup algebra of Z n . For a Laurentpolynomial f = Í α ∈ Z n c α x α , its Newton polytope is the integerpolytope generated by the set of the exponents α of the non-zerocoefficients of f ; that is, NP ( f ) : = Convex Hull ({ α ∈ Z n , c α , }) .Instead of working over K [ Z n ] , we embed f in a subalgebrarelated to its Newton polytope, given by K [C NP ( f ) ∩ Z n ] . In thisway we exploit the sparsity of the (polynomials of the) system. Definition 2.6 (Semigroup algebra of polytopes).
We consider r integer polytopes ∆ , . . . , ∆ r ⊂ R n such that their Minkowski sum, ∆ : = Í ri = ∆ i , has dimension n and is its vertex; in particular, as a vertex of every Newton polytope ∆ i . We also consider thepolytope ¯ ∆ : = Í ( ∆ i × { e i }) , which is the Cayley embedding of ∆ , . . . , ∆ r .In what follows, we work with the semigroup algebras K [ S ∆ ] : = K [C ∆ ∩ Z n ] and K [ S h ∆ ] : = K [C ¯ ∆ ∩ Z n + r ] . We will write the mono-mials in K [ S h ∆ ] as X ( α , d ) , where α ∈ (C ∆ ∩ Z n ) and d ∈ N r . The algebra K [ S h ∆ ] is N r -multigraded as follows: for every d = ( d , . . . , d r ) ∈ N r , K [ S h ∆ ] d is the K -vector space spanned by themonomials { X ( α , d ) : α ∈ ( Í d i · ∆ i )∩ Z n } . Then, F ∈ K [ S h ∆ ] d is ho-mogeneous and has multidegree d , which we denote by mdeg ( F ) .We can think K [ S ∆ ] as the “dehomogenization” of K [ S h ∆ ] . Definition 2.7 (Dehomogenization morphism).
The dehomogeniza-tion morphism from K [ S h ∆ ] to K [ S ∆ ] is the surjective ring homo-morphism χ : K [ S h ∆ ] → K [ S ∆ ] that maps the monomials X ( α , d ) ∈ K [ S h ∆ ] to χ ( X ( α , d ) ) : = X α ∈ K [ S ∆ ] .If L is a set of homogeneous polynomials in K [ S h ∆ ] , then weconsider χ (L) = { χ ( G ) : G ∈ L} . Observation 2.8. As is a vertex of ∆ , there is a monomial X ( , e i ) ∈ K [ S h ∆ ] , for every i ∈ [ r ] . Hence, given a finite set of mono-mials X α , . . . , X α k ∈ K [ S ∆ ] , we can find a multidegree d ∈ N r such that X ( α , d ) , . . . , X ( α k , d ) ∈ K [ S ∆ ] d .Given a system of polynomials f , . . . , f m ∈ K [ S ∆ ] , we can finda multidegree d ∈ N r and homogeneous polynomials F , . . . , F m ∈ K [ S h ∆ ] d so that it holds χ ( F i ) = f i , for every i ∈ [ m ] .Moreover, given homogeneous polynomials F , . . . , F m ∈ K [ S h ∆ ] and an affine polynomial д ∈ h χ ( F ) , . . . , χ ( F m )i , there is an homo-geneous polynomial G ∈ h F , . . . , F m i such that χ ( G ) = д . Observation 2.9.
If we fix a multidegree d ∈ N r , then the map χ restricted to K [ S h ∆ ] d is injective. We recall some definitions related to Groebner basis over semi-group algebras from [20]. Let S be a pointed affine semigroup. Definition 2.10 (Monomial order).
Given a pointed semigroup al-gebra K [ S ] , a monomial order for K [ S ] , say < , is a total order forthe monomials in K [ S ] such that: • For any α ∈ S \ { } , it holds X < X α . • For every α , β , γ ∈ S , if X α < X β then X α + γ < X β + γ . Observation 2.11.
Monomial orders always exist for pointed affinesemigroups. To construct them, first we embed any pointed affinesemigroup of dimension n in a pointed rational cone C ⊂ R n . Then,we choose n linearly independent forms l , . . . , l n from the dual coneof C , which is { l : R n → R | ∀ α ∈ C , l ( α ) ≥ } . We define themonomial order so that X α < X β if and only if there is a k ≤ n such that for all i < k it holds l i ( α ) = l i ( β ) and l k ( α ) < l k ( β ) .Definition 2.12 (Leading monomial). Given a monomial order < for a pointed affine semigroup algebra K [ S ] and a polynomial f ∈ K [ S ] , its leading monomial, LM < ( f ) is the biggest monomial of f with respect to the monomial order < .The exponent of the leading monomial of f always correspondsto a vertex of NP ( f ) . Definition 2.13 (Gröbner basis).
Let K [ S ] be a pointed affine semi-group algebra and consider a monomial order < for K [ S ] . For anideal I ⊂ K [ S ] , a set G ⊂ I is a Gröbner basis of I if { LM < ( д ) : д ∈ G } generates the same ideal as { LM < ( f ) : f ∈ I } .In other words, if for every f ∈ I , there is д ∈ G and X α ∈ K [ S ] such that LM < ( f ) = X α LM < ( д ) . atías R. Bender, Jean-Charles Faugère, and Elias Tsigaridas As S is finitely generated, the algebra K [ S ] is a Noetherian ring[24, Thm. 7.7]. Hence, for any monomial order and any ideal, thereis always a finite Gröbner basis.We will consider monomial orders for K [ S h ∆ ] that we can relateto monomial orders in K [ S ∆ ] and K [ N r ] . Definition 2.14 (Multigraded monomial order).
We say that a mono-mial order < for K [ S h ∆ ] is multigraded, if there are monomial orders < ∆ for K [ S ∆ ] and < h for K [ N r ] such that, for every X ( α , d ) , X ( α , d ) ∈ K [ S h ∆ ] , it holds X ( α , d ) < X ( α , d ) ⇐⇒ ( X d < h X d or d = d and X α < ∆ X α . (1)Multigraded monomial orders are “compatible” with the deho-mogenization morphism (Def. 2.7). Remark 2.15.
In what follows, given a multigraded monomial or-der < for K [ S h ∆ ] , we also use the same symbol, that is < , for the asso-ciated monomial order of K [ S ∆ ] . Lemma 2.16.
Consider a polynomial f ∈ K [ S ∆ ] . Let < be a multi-graded monomial order. For any multidegree d and any homogeneous F ∈ K [ S h ∆ ] d such that χ ( F ) = f , it holds LM < ( f ) = χ ( LM < ( F )) . The Bernstein-Kushnirenko-Khovanskii (BKK) theorem bounds the(finite) number of solutions of a square system of sparse Laurentpolynomials over the torus ( C ∗ ) n , where C ∗ : = C \ { } . Definition 2.17 (Mixed volume).
Let ∆ , . . . , ∆ n ∈ R n be integerpolytopes. Their mixed volume, MV ( ∆ , . . . , ∆ n ) , is the alternatingsum of the number of integer points of the polytopes obtained byall possible Minkowski sums, that is MV ( ∆ , . . . , ∆ n ) = (− ) n + n Õ k = (− ) n − k (cid:16) Õ I ⊂{ ,..., n } I = k (cid:16) ( ∆ I + · · · + ∆ Ik ) ∩ Z n (cid:17) (cid:17) . (2) Theorem 2.18 (BKK bound [9, Thm 7.5.4]).
Let f , . . . , f n be asystem of polynomials with Newton polytopes ∆ , . . . , ∆ n having afinite number of solutions over ( C ∗ ) n . The mixed volume MV ( ∆ , . . . , ∆ n ) upper bounds the number of solutions of the system over the torus ( C ∗ ) n . If the non-zero coefficients of the polynomials are generic, thenthe bound is tight. Toric varieties relate semigroup algebras with the torus ( C ∗ ) n .A toric variety is an irreducible variety X that contains ( C ∗ ) n asan open subset such that the action of ( C ∗ ) n on itself extends to analgebraic action of ( C ∗ ) n on X [10, Def. 3.1.1]. Semigroup algebrascorrespond to the coordinate rings of the affine pieces of X .Given an integer polytope ∆ , we can define a projective com-plete normal irreducible toric variety X associated to it [10, Sec. 2.3].Likewise, given a polynomial system ( f , . . . , f m ) , we can definea projective toric variety X associated to the Minkowski sum oftheir Newton polytopes. We can homogenize these polynomialsin a way that they belong to the total coordinate ring of X [10,Sec. 5.4]. This homogenization is related to the facets of the poly-topes.To be more precise, given an integer polytope ∆ ⊂ R n , we saythat an integer polytope ∆ is a N -Minkowski summand of ∆ if there is a k ∈ N and another polytope ∆ such that ∆ + ∆ = k · ∆ [10, Def. 6.2.11]. Every N -Minkowski summand ∆ of ∆ definesa torus-invariant basepoint free Cartier divisor D of the projectivetoric variety X associated to ∆ [10, Cor. 6.2.15]. This divisor definesan invertible sheaf O X ( D ) whose global sections form the vectorspace of polynomials in K [ Z n ] whose Newton polytopes are con-tained in ∆ [32, Lem. 1]. Therefore, to homogenize f , . . . , f m over X we need to choose polytopes ∆ , . . . , ∆ m such that all of themare N -Minkowski summands of ∆ associated to X and NP ( f i ) ⊂ ∆ i .Hence, for any homogeneous F ∈ K [ S h ∆ ] d , we can homogenize χ ( F ) with respect to the N -Minkowski summand Í i d i ∆ i of ∆ .We alert the reader that homogeneity in K [ S h ∆ ] d is differentfrom homogeneity in the total coordinate ring of X , see [10, Sec. 5.4]but they are related through the degree d . Definition 2.19 (Solutions at infinity).
Let ( f , . . . , f m ) be a sys-tem of polynomials. Let X be the projective toric variety associ-ated to a polytope ∆ such that the Newton polytope of f i is a N -Minkowski summand of ∆ , for all i . We say that the system hasno solutions at infinity with respect to X if the homogenized sys-tem with respect to their Newton polytopes has no solutions over X \ ( C ∗ ) n . Proposition 2.20 ([32, Thm. 3]).
Consider a system ( f , . . . , f n ) having finite number of solutions over ( C ∗ ) n . Let X be the projectivetoric variety associated to the corresponding Newton polytopes. Then,the number of solutions of the homogenized system over X , countingmultiplicities, is exactly the BKK bound. When the original systemhas no solutions at infinity, then the BKK is tight over ( C ∗ ) n ⊂ X .Definition 2.21 (Koszul complex, [12, Sec. 17.2]). For a sequence ofhomogeneous F , . . . , F k ∈ K [ S h ∆ ] of multidegrees d , . . . , d k anda multidegree d ∈ N r , we denote by K( F , . . . , F k ) d the strand ofthe Koszul complex of F , . . . , F k of multidegree d , that is, K( F , . . . , F k ) d : 0 → (K k ) d δ k −−→ . . . δ −−→ (K ) d → , where, for ≤ t ≤ k , we have (K t ) d : = Ê I ⊂{ ,..., k } I = t K [ S h ∆ ] ( d − Í i ∈ I d i ) ⊗ ( e I ∧ · · · ∧ e I t ) . The maps (differentials) act as follows: δ t (cid:16) Õ I ⊂{ ,..., k } I = t д I ⊗ ( e I ∧ · · · ∧ e I t ) (cid:17) = Õ I ⊂{ ,..., k } I = t t Õ i = (− ) i − F I i д I ⊗ ( e I ∧ · · · ∧ c e I i ∧ · · · ∧ e I t ) . (3) The expression ( e I ∧· · ·∧ c e I i ∧· · ·∧ e I t ) denotes that we skip theterm e I i from the wedge product. We denote by H t ( F , . . . , F k ) d the t -th Koszul homology of K( F , . . . , F k ) d , that is H t ( F , . . . , F k ) d : = ( ker ( δ t )/ im ( δ t + )) d . The -th Koszul homology is H ( F , . . . , F k ) (cid:27) ( K [ S h ∆ ]/h F , . . . , F k i) . Definition 2.22 (Koszul and sparse regularity).
A sequence F , . . . , F k ∈ K [ S h ∆ ] is Koszul regular if for every d ∈ N r coordinate-wise greater than or equal to D k : = Í ki = d i , that is, d ≥ D k , andfor every t > , the t -th Koszul homology vanishes at degree d , röbner Basis over Semigroup Algebras that is H t ( F , . . . , F k ) d = . We say that the sequence is (sparse)regular if F , . . . , F j is Koszul regular, for every j ≤ k . Observation 2.23.
Note that Koszul regularity does not dependon the order of the polynomials, as (sparse) regularity does.
Kushnirenko’s proof of the BKK bound [29, Thm. 2] followsfrom Koszul regularity.
To compute Gröbner basis over K [ S ∆ ] we work over K [ S h ∆ ] . Wefollow the classical approach of Lazard [30] adapted to the semi-group case, see also [20]; we “linearize” the problem by reducingthe Gröbner basis computation to a linear algebra problem. Lemma 3.1.
Consider F , . . . , F m ∈ K [ S h ∆ ] and a multigradedmonomial order < for K [ S ∆ ] (Def. 2.14). There is a multidegree d and homogeneous { G , . . . , G t } ⊂ h F , . . . , F m i ∩ K [ S h ∆ ] d such that { χ ( G ) , . . . , χ ( G t )} is a Gröbner basis of the ideal h χ ( F ) , . . . , χ ( F m )i with respect to the associated monomial order < (Rem. 2.15). Proof.
Let д , . . . , д t ∈ K [ S ∆ ] be a Gröbner basis for the ideal h χ ( F ) , . . . , χ ( F m )i with respect to < . By Obs. 2.8, there are poly-nomials ¯ G , . . . , ¯ G t ∈ h F , . . . , F m i such that χ ( ¯ G i ) = д i , for i ∈ [ t ] .Consider d ∈ N r such that d ≥ mdeg ( ¯ G i ) , for i ∈ [ t ] . It suffices toconsider G i = X ( , d − mdeg ( ¯ G i )) ¯ G i ∈ K [ S h ∆ ] d , for i ∈ [ t ] . (cid:3) When we know a multidegree d that satisfies Lem. 3.1, we cancompute the Gröbner basis over K [ S ∆ ] using linear algebra. Forthis task we need to introduce the Macaulay matrix. Definition 3.2 (Macaulay matrix).
A Macaulay matrix M of de-gree d ∈ N r with respect to a monomial order < is a matrix whosecolumns are indexed by all monomials X ( α , d ) ∈ K [ S h ∆ ] d and therows by polynomials in K [ S h ∆ ] d . The indices of the columns aresorted in decreasing order with respect to < . The element of M whose row corresponds to a polynomial F and whose column cor-responds to a monomial X ( α , d ) is the coefficient of the monomial X ( α , d ) of F . Let Rows (M) be the set of non-zero polynomials thatindex the rows of M and LM < ( Rows (M)) be the set of leadingmonomials of these polynomials.
Remark 3.3.
As the columns of the Macaulay matrices are sortedin decreasing order with respect to a monomial order, the leadingmonomial of a polynomial associated to a row corresponds to the in-dex of the column of the first non-zero element in this row.Definition 3.4.
Given a Macaulay matrix M , let f M be a newMacaulay matrix corresponding to the row echelon form of M .We can compute f M by applying Gaussian elimination to M . Remark 3.5.
When we perform row operations (excluding mul-tiplication by 0) to a Macaulay matrix, we do not change the idealspanned by the polynomials corresponding to its rows.
We use Macaulay matrices to compute a basis for the vectorspace h F , . . . , F k i d : = h F , . . . , F k i ∩ K [ S h ∆ ] d by Gaussian elimi-nation. Lemma 3.6.
Consider homogeneous polynomials F , . . . , F k ∈ K [ S h ∆ ] of multidegrees d , . . . , d k and a multigraded monomial order < . Let Algorithm 1
ComputeGB
Input: f , . . . , f k ∈ K [ S ∆ ] , a monomial order < . Output:
Gröbner basis for h f , . . . , f k i with respect to < .1: for all f i do
2: Choose F i ∈ K [ S h ∆ ] di of multidegree d i such that χ ( F i ) = f i .3: Pick a big enough d ∈ N r that satisfies Lem. 3.1.4: M k d ← Macaulay matrix of multidegree d with respect to a multigradedmonomial order associated to < .5: for all F i do for all X ( α , d − d i ) ∈ K [ S h ∆ ] d − di do
7: Add the polynomial X ( α , d − d i ) F i as row to M k d .8: f M k d ← GaussianElimination ( M k d )9: return χ ( Rows ( f M k d )) M k d be the Macaulay matrix whose rows correspond to the polyno-mials that we obtain by considering the product of every monomialof multidegree d − d i and every polynomial F i ; that is Rows (M k d ) = n X ( α , d − d i ) F i : i ∈ [ k ] , X ( α , d − d i ) ∈ K [ S h ∆ ] d − d i o . (4) Let f M k d be the row echelon form of the Macaulay matrix M k d (Def. 3.4).Then, the set of the leading monomials of the polynomials in Rows ( f M k d ) with respect to < is the set of all the leading monomials of the ideal h F , . . . , F k i at degree d . Proof.
We prove that LM < ( Rows ( f M k d )) = LM < (h F , . . . , F k i d ) .First, we show that LM < ( Rows ( f M k d )) ⊇ LM < (h F , . . . , F k i d ) . Let G be a polynomial in the vector space of polynomials of degree d in h F , . . . , F k i . This vector space, h F , . . . , F k i d , is isomorphic to therow space of M k d , which, in turn, is the same as the row space of f M k d , by Rem. 3.5. Hence, there is a vector v in the row space of f M k d that corresponds to G . Let s be the index of the first non-zero ele-ment of v . As f M k d is in row echelon form and v belongs to its rowspace, there is a row of f M k d such that its first non-zero element isalso at the s -th position. Let F be the polynomial that correspondsto this row. Finally, the leading monomials of the polynomials F and G are the same, that is LM < ( G ) = LM < ( F ) , by Rem. 3.3.The other direction is straightforward. (cid:3) Theorem 3.7.
Consider the ideal generated by homogeneous poly-nomials F , . . . , F k ∈ K [ S h ∆ ] of multidegrees d , . . . , d k . Consider amultigraded monomial order < and a multidegree d ∈ N r that sat-isfy Lem. 3.1. Let M k d and f M k d be the Macaulay matrices of Lem. 3.6.Then, the set χ ( Rows ( f M k d )) , see Def. 2.7, contains a Gröbner basisof the ideal h χ ( F ) , . . . , χ ( F k )i ⊂ K [ S ∆ ] with respect to < . Proof.
Let R : = Rows ( f M k d ) be the set of polynomials indexingthe rows of f M k d . By Lem. 3.6, for every G ∈ h F , . . . , F k i d there is a F ∈ R such that LM < ( G ) = LM < ( F ) . As < is a multigraded order, itholds LM < ( χ ( G )) = LM < ( χ ( F )) (Lem. 2.16). As d satisfies Lem. 3.1for every h ∈ h χ ( F ) , . . . , χ ( F k )i there is G ∈ h F , . . . , F k i d suchthat LM < ( χ ( G )) divides LM < ( h ) . Hence, there is an F ∈ R suchthat LM < ( χ ( F )) divides LM < ( h ) . Therefore, R is a Gröbner basisfor h χ ( F ) , . . . , χ ( F k )i . (cid:3) Theorem 3.7 leads to an algorithm for computing Gröbner basesthrough a Macaulay matrix and Gaussian elimination. atías R. Bender, Jean-Charles Faugère, and Elias Tsigaridas
Algorithm 2
ReduceMacaulay
Input:
Homogeneous F , . . . , F k ∈ K [ S h ∆ ] of multidegree d , . . . , d k , a multide-gree d , and a monomial order < . Output:
The Macaulay matrix of h F , . . . , F k i d ∈ K [ S h ∆ ] with respect to < in rowechelon form.1: M k d ← Macaulay matrix with columns indexed by the monomials in K [ S h ∆ ] d in decreasing order wrt < if k > then f M k − d ← ReduceMacaulay ({ F , . . . , F k − } , d , < ) f M k − d − d k ← ReduceMacaulay ({ F , . . . , F k − } , d − d k , < ) for F ∈ Rows ( f M k − d ) do
6: Add the polynomial F as a row to M k d .7: for X ( α , d − d k ) ∈ K [ S h ∆ ] d − d k \ LM < ( Rows ( f M k − d − d k )) do
8: Add the polynomial X ( α , d − d k ) F k as a row to M k d .9: f M k d ← GaussianElimination ( M k d )10: return f M k d If we consider all the polynomials of the set in Eq. (4), then manyof them are linearly dependent. Hence, when we construct theMacaulay matrix of Thm. 3.7 and perform Gaussian elimination,many of the rows reduce to zero; this forces Alg. 1 to perform un-necessary computations. We will extend to F5 criterion [16] in oursetting to avoid redundant computations.
Theorem 3.8 (Koszul F5 criterion).
Consider homogeneous poly-nomials F , . . . , F k ∈ K [ S h ∆ ] of multidegrees d , . . . , d k and a multi-degree d ∈ N r such that d ≥ d k , that is coordinate-wise greater thanor equal to d k . Let M k − d and M k − d − d k be the Macaulay matrices ofdegrees d and d − d k , respectively, of the polynomials F , . . . , F k − as in Thm. 3.7, and let f M k − d and f M k − d − d k be their row echelon forms.For every X ( α , d − d k ) ∈ LM < ( Rows ( f M k − d − d k )) , the polynomial X ( α , d − d k ) F k is a linear combination of the polynomials Rows ( f M k − d ) ∪ (cid:26) X ( β , d − d k ) F k : X ( β , d − d k ) ∈ K [ S h ∆ ] d − d k and X ( β , d − d k ) < X ( α , d − d k ) (cid:27) . Proof. If X ( α , d − d k ) ∈ LM < ( Rows ( f M k − d − d k )) , then there is G ∈ K [ S h ∆ ] d − d k such that X ( α , d − d k ) + G ∈ h F , . . . , F k − i d − d k and X ( α , d − d k ) > LM < ( G ) . So, there are homogeneous H , . . . , H k − ∈ K [ S h ∆ ] such that X ( α , d − d k ) + G = Í i H i F i . The proof follows bynoticing that X ( α , d − d k ) F k = Í k − i = ( F k H i ) F i − G F k . (cid:3) In the following, M k d is not the Macaulay matrix of 3.6. It con-tains less rows because of the Koszul F5 criterion. However, bothmatrices have the same row space, so we use the same name. Corollary 3.9.
Using the notation of Thm. 3.8, let M k d be a Macaulaymatrix of degree d wrt the order < whose rows are Rows ( f M k − d ) ∪ ( X ( β , d − d k ) F k : X ( β , d − d k ) ∈ K [ S h ∆ ] d − d k and X ( β , d − d k ) < LM < ( Rows ( f M k − d − d k )) ) The row space of M k d and the Macaulay matrix of Lem. 3.6 are equal. The correctness of Alg. 2 follows from Thm. 3.8.
Lemma 3.10. If H ( F , . . . , F k ) d = and there is a syzygy Í i G i F i = such that G i ∈ K [ S h ∆ ] d − d i , then G k ∈ h F , . . . , F k − i d − d k . Proof.
We consider the Koszul complex K( F , . . . , F k ) (Def. 2.21).As Í i G i F i = δ ( G , . . . , G k ) , the vector of polynomials ( G , . . . , G k ) belongs to the Kernel of δ . As H ( F , . . . , F k ) d vanishes, the ker-nel of δ is generated by the image of δ . The latter map is ( H , , . . . , H k − , k ) 7→ Õ ≤ i < j ≤ k H i , j ( F j e i − F i e j ) , where e i and e j are canonical basis of R k . Hence, there are homo-geneous polynomials ( H , , . . . , H k − , k ) such that ( G , . . . , G k ) = Õ ≤ i < j ≤ k H i , j ( F j e i − F i e j ) . Thus, G k = Í k − i = H i , k F i and so G k ∈ h F , . . . , F k − i d − d k . (cid:3) The next lemma shows that we avoid all redundant computations,that is all the rows reducing to zero during Gaussian elimination.
Lemma 3.11. If H ( F , . . . , F k ) d = , then all the rows of thematrix M k d in Alg. 2 are linearly independent. Proof.
By construction, the rows of M k d corresponding to f M k − d are linearly independent because the matrix is in row echelon form.Hence, if there are rows that are not linearly independent, then atleast one of them corresponds to a polynomial of the form X ( α , d − d k ) F k .The right action of the Macaulay matrix M k d represents a mapequivalent to the map δ from the strand of Koszul complex K( F , . . . , F k ) d . Hence, if some of the rows of the matrix are lin-early dependent, then there is an element in the kernel of δ . Thatis, there are G i ∈ K [ S h ∆ ] d − d i such that • Í k − i = G i F i belongs to the linear span of Rows ( f M k − d ) , • the monomials of G k do not belong to LM < ( Rows ( f M k − d − d k )) , and • Í ki = G i F i = .By Lem. 3.10, G k ∈ h F , . . . , F k − i d − d k . By Lem. 3.6 and Cor. 3.9,the leading monomials of Rows ( f M k − d − d k ) and the ideal h F , . . . , F k − i at degree d − d k are the same. Hence, we reach a contradiction be-cause we have assumed that the leading monomial of G k does notbelong to LM < ( Rows ( f M k − d − d k )) . (cid:3) Corollary 3.12. If F , . . . , F k is a sparse regular polynomial sys-tem (Def. 2.22) and d ∈ N r is such that d ≥ ( Í i d i ) , then ReduceMacaulay ( F , . . . , F k , d , < ) only considers matrices with lin-early independent rows and avoids all redundant computations. To benefit from the Koszul F5 criterion and compute with smallermatrices during the Gröbner basis computation we should replaceLines 4 – 8 in Alg. 1 by
ReduceMacaulay ( F , . . . , F k , d , < ) (Alg. 2). -DIM SYSTEMS We introduce an algorithm, that takes as input a -dimensionalideal I and computes a Gröbner basis for the ideal (cid:16) I : h Î j x j i ∞ (cid:17) .The latter corresponds to the ideal associated to the intersection ofthe torus ( C ∗ ) n with the variety defined by I .Let f , . . . , f n ∈ K [ x ] be a square -dimensional system. Firstwe embed each f i in K [ Z n ] . We multiply each polynomial by anappropriate monomial, X β i ∈ K [ Z n ] , so that is a vertex of each röbner Basis over Semigroup Algebras new polynomial, as well as, a vertex of their Minkowski sum. Letthe Newton polytopes be ∆ i = NP ( X β i f i ) , for ≤ i ≤ n , Let ∆ be the standard n -simplex; it is the Newton polytope of NP ( + Í i x i ) . We consider the algebras K [ S ∆ ] and K [ S h ∆ ] associated tothe polytopes ∆ , . . . , ∆ n and the embedding X β f , . . . , X β f n ∈ K [ S ∆ ] . For each i , we consider F i ∈ K [ S h ∆ ] e i such that χ ( F i ) = X β i f i ∈ K [ S ∆ ] . Assumption 4.1.
Using the previous notation, let X be the projec-tive toric variety associated to ∆ + · · · + ∆ n (see also the discussion ontoric varieties at Sec. 2.3). Assume that the system ( f , . . . , f n ) has nosolutions at infinity with respect to X (Def. 2.19). Further, assume thatthe system ( f , f , . . . , f n ) , where f is generic linear polynomial, hasno solutions over ( C ∗ ) n . Lemma 4.2 ([32, Thm. 3.a]).
Under Assum. 4.1, for every d ∈ N n + such that d ≥ Í i > e i , it holds H ( F , . . . , F n ) d (cid:27) K [ Z n ]/h f , . . . , f n i . Lemma 4.3 ([32, Thm. 3.c]).
Under Assum. 4.1, for every homoge-neous polynomial F ∈ K [ S h ∆ ] d such that the system ( f , . . . , f n , χ ( F )) has no solutions over ( C ∗ ) n , the system ( F , . . . , F n , F ) is Koszul reg-ular (Def. 2.22) and, for every d ∈ N n + such that d ≥ Í i e i + d , h F , . . . , F n , F i d = K [ S h ∆ ] d . Proof.
The homogenization of system ( f , . . . , f n , χ ( F )) withrespect to the toric variety X has no solutions over X (see at Sec. 2.3the discussion before Def. 2.19). To see this, notice that by As-sum. 4.1 the homogenization of the system ( f , . . . , f n ) with re-spect to X has no solutions over X \ ( C ∗ ) n (see also Def. 2.19).Moreover, we also assume that there are no solutions over ( C ∗ ) n of ( f , . . . , f n , χ ( F )) .Now, the proof follows from the argument in the proof of [32,Thm. 3]. This argument is the same as in [23, Prop. 3.4.1], wherethe stably twisted condition is given by [32, Thm. 1]. (cid:3) Corollary 4.4.
For any monomial X ( α , D e ) ∈ K [ S h ∆ ] D e , thesystem ( F , . . . , F n , X ( α , D e ) ) is Koszul regular. For every d ∈ N n + such that d ≥ Í i e i + D e , it holds h F . . . F n , X ( α , D e ) i d = K [ S h ∆ ] d . Fix a graded monomial order < for K [ S h ∆ ] ; L is the set of monomialsthat are not leading monomials of h F , . . . , F n i Í i ≥ e i , that is L : = (cid:26) X ( α , Í i ≥ e i ) ∈ K [ S h ∆ ] Í i ≥ e i : ∀ G ∈ h F , . . . , F n i Í i ≥ e i , LM < ( G ) , X ( α , Í i ≥ e i ) (cid:27) We will prove that the dehomogenization of these monomials, χ (L) , forms a monomial basis for K [ Z n ]/h f , . . . , f n i . Lemma 4.5.
The monomials in the set χ (L) are K -linearly inde-pendent in K [ Z n ]/h f , . . . , f n i . Proof.
Assume that the lemma does not hold. Hence, there are c , . . . , c v ∈ K , not all of them , and д , . . . , д n ∈ K [ Z n ] such that Í i c i χ (L i ) = Í i д i f i . We can clear the denominators, introducedby the д i ’s, by choosing a monomial X α ∈ K [ N n ] such that, forevery i , (cid:16) X α X β i д i (cid:17) ∈ K [ N n ] . Moreover, there is a degree D ∈ N andhomogeneous polynomials G i ∈ K [ S h ∆ ] of multidegrees ( D e + Í j > e j − e i ) such that χ ( G i ) = (cid:16) X α X β i д i (cid:17) and X ( α , D e ) Í i c i L i = Í i G i F i . By Lem. 4.3, ( F , . . . , F n , X ( α , D e ) ) is Koszul regular andso, by Lem. 3.10, Í i c i L i ∈ h F , . . . , F n i Í i > e i . So, a monomial in L is a leading monomial of an element in h F , . . . , F n i Í i > e i . Thisis a contradiction as, by construction, there is no monomial in L which is a leading monomial of a polynomial in h F , . . . , F n i Í i > e i . (cid:3) Corollary 4.6.
The set of monomials χ (L) is a monomial basisof K [ Z n ]/h f , . . . , f n i . Proof.
By Lem. 4.2, the number of elements in the set L andthe dimension of K [ Z n ]/h f , . . . , f n i is the same. By Obs. 2.9, thesets L and χ (L) have the same number of elements. By Lem. 4.5,the monomials in the set χ (L) are linearly independent. (cid:3) Remark 4.7.
One way to compute the set L is to compute a basisof the vector space h F , . . . , F n i Í i ≥ e i using Alg. 2, that is ReduceMacaulay (cid:0) ( F , . . . , F n ) , Í i ≥ e i , < (cid:1) . For each F ∈ K [ S h ∆ ] e , we will construct a Macaulay matrix for ( F , . . . , F n , F ) at multidegree : = Í i e i , say M( F ) ; from this ma-trix we will recover the multiplication map of χ ( F ) in K [ x ]/h f , . . . , f n i .The rows of M( F ) are of two kinds: • the polynomials in Rows ( f M n ) , where f M n = ReduceMacaulay (( F , . . . , F n ) , , < ) , • the polynomials of the form m F , where m ∈ L . Lemma 4.8.
The matrix M( F ) is always square. It is full-rank ifand only if ( F , . . . , F N , F ) is Koszul regular. Proof.
According to the Koszul F5 criterion (see Thm. 3.8), therow space spanned by M( F ) is the same as the vector space h F , . . . , F n , F i for any choice of F . We can consider an F suchthat ( F , . . . , F n , F ) is Koszul regular, by Cor. 4.4. Then, the rows of M( F ) generate K [ S h ∆ ] and, by Lem. 3.11, the rows of M( F ) arelinearly independent. Hence, by Lem. 4.3, for this particular F , thematrix M( F ) is square and full-rank. However, the matrix M( F ) is square for any choice of F , because its number of rows does notdepend on F . Nevertheless, it is not full-rank for any choice of F . If M( F ) is full-rank, then ( F , . . . , F n , F ) is Koszul regular be-cause, by the sparse Nullstellensatz [38, Thm. 2], the homogeniza-tion of the system ( f , . . . , f n , χ ( F )) has no solutions over ( C ∗ ) n .Consequently, the proof follows from Lem. 4.3. (cid:3) We reorder the columns of M( F ) as shown in Eq. (5), such that • the columns of the submatrix (cid:16) M , ( F ) M , ( F ) (cid:17) correspond to mono-mials of the form m X ( , e ) , where m ∈ L , and • the rows of ( M , ( F ) | M , ( F ) ) are polynomials of the form m F , where m ∈ L . Rows ( f M n ) F · L M , ( F ) M , ( F ) X ( , ) ·L z }| { M , ( F ) M , ( F ) (5) We prove that M , ( F ) is invertible and the Schur complement of M , ( F ) , M c , ( F ) : = ( M , − M , M − , M , )( F ) , is the multipli-cation map of χ ( F ) in the basis χ (L) of K [ Z n ]/h f , . . . , f n i . atías R. Bender, Jean-Charles Faugère, and Elias Tsigaridas Lemma 4.9. If ( F , . . . , F n , X ( , e ) ) is Koszul regular then, for any F ∈ K [ S h ∆ ] e , the matrix M , ( F ) is invertible. Proof.
By Lem. 4.8, as the system ( F , . . . , F n , X ( , e ) ) is Koszulregular, then the matrix M( X ( , e ) ) is invertible. As M , ( X ( , e ) ) is the zero matrix and M , ( X ( , e ) ) is the identity,then M , ( X ( , e ) ) must be invertible. By construction, the matrices M , ( F ) and M , ( F ) are independent of the choice of F . Hence,for any F the matrix M , ( F ) is invertible. (cid:3) Theorem 4.10.
The multiplication map of χ ( F ) in the mono-mial basis χ (L) of K [ Z n ]/h f , . . . , f n i is the Schur complement of M , ( F ) , that is M c , ( F ) : = ( M , − M , M − , M , )( F ) . Proof.
Note that for every F ∈ K [ S h ∆ ] e and each element L i of L , L i F ≡ X ( , e ) Í j ( M c , ( F )) i , j L j in K [ S h ∆ ]/h F , . . . , F n i , where ( M c , ( F ) i , j is the ( i , j ) element of the matrix M c , ( F )) . Hence,if we dehomogenize this relation we obtain that, χ ( L i ) χ ( F ) ≡ Í j ( M c , ( F )) i , j χ ( L j ) in K [ S ∆ ]/h X β f , . . . , X β n f n i . As K [ S ∆ ] ⊂ K [ Z n ] , the same relation holds in K [ Z n ]/h f , . . . , f n i . By Cor. 4.6,the set χ (L) is a monomial basis of K [ Z n ]/h f , . . . , f n i . Therefore, M c , ( F ) is the multiplication map of F in K [ Z n ]/h f , . . . , f n i . (cid:3) Using the multiplication maps in K [ Z n ]/h f , . . . , f n i and theFGLM algorithm [17], we can compute a Gröbner basis for h f , . . . , f n i : h Î i x i i ∞ over K [ x ] . The latter is the saturation over K [ N n ] of the ideal h f , . . . , f m i by the product of all the variables. Lemma 4.11.
Consider polynomials f , . . . , f m ⊂ K [ N n ] suchtheir ideal over K [ Z n ] , h f , . . . , f m i K [ Z n ] , is 0-dimensional. Let h f , . . . , f m i K [ N n ] be the ideal generated by f , . . . , f m over K [ N n ] .Then, the sets h f , . . . , f m i K [ Z n ] ∩ K [ N n ] and h f , . . . , f m i K [ N n ] : h Î i x i i ∞ are the same. The latter is an ideal over K [ N n ] . Proof.
Consider f ∈ h f , . . . , f m i K [ Z n ] . Then there are д i ∈ K [ Z n ] such that f = Í i д i f i . We can clear the denominators intro-duced by the д i ’s by multiplying both sides by a monomial ( Î j x j ) d ,where d is big enough. Then, ( Î j x j ) d f = Í i (( Î j x j ) d д i ) f i and (( Î j x j ) d д i ) ∈ K [ N n ] . Thus, ( Î j x j ) d f ∈ h f , . . . , f m i and f ∈h f , . . . , f m i : h Î j x j i ∞ . The opposite direction is straightforwardas Î i x i is a unit in K [ Z n ] . (cid:3) We can perform FGLM over K [ Z n ] to recover a Gröbner basisfor h f , . . . , f m i K [ N n ] : h Î i x i i ∞ by considering the multiplicationmaps of each x i . These correspond to M c , ( X ( α i , e ) ) , where α i issuch that χ ( X ( α i , e ) ) = x i . We skip the details of this procedure. We estimate the arithmetic complexity of the algorithm in Sec 4;it is polynomial with respect to the Minkowski sum of the poly-topes. We omit the cost of computing all the monomials in K [ S h ∆ ] d and we only consider the complexity to read them. Our purposeis to highlight the dependency on the Newton polytopes. A moredetailed analysis might give sharper bounds. Definition 4.12.
For polytopes ∆ , . . . , ∆ n and for each multide-gree d ∈ N n + of K [ S h ∆ ] , let P ( d ) be the number of integer pointsin the Minkowski sum of the polytopes given by d , P ( d ) = (cid:16) ( Í nj = d j ∆ j ) ∩ Z n (cid:17) . Note that P ( d ) equals the number of different monomials in K [ S h ∆ ] d . Lemma 4.13.
Let F , . . . , F k ∈ K [ S h ∆ ] be a (sparse) regular se-quence and let d i ∈ N n + be the multidegree of F i , for i ∈ [ k ] .Consider a multigraded monomial order < . For every multidegree d ∈ N n + such that d ≥ Í i d i , the arithmetic complexity of comput-ing ReduceMacaulay (( F , . . . , F k ) , d , < ) is O ( k + P ( d ) ω ) , where ω is the constant of matrix multiplication. Proof.
By Cor. 3.12, as F , . . . , F k ∈ K [ S h ∆ ] is a (sparse) regularsequence and d ≥ Í i d i , all the matrices that appear during thecomputations of ReduceMacaulay (( F , . . . , F k ) , d , < ) are full-rankand their rows are linearly independent. Hence, their number ofrows is at most their number of columns. The number of columnsof a Macaulay matrix of multidegree d is P ( d ) . Thus, in this case,the complexity of Gaussian elimination is O ( P ( d ) ω ) [45]. If C ( k , d ) is cost of ReduceMacaulay (( F , . . . , F k ) , d , < ) , then we have thefollowing recursive relation C ( k , d ) = (cid:26) O ( P ( d ) ω ) if k = , C ( k − , d ) + C ( k − , d − d k ) + O ( P ( d ) ω ) if k > .The cost C ( k − , d ) is greater than C ( k − , d − d k ) , as it involvesbigger matrices. Hence, we obtain C ( k , d ) = O ( k + P ( d ) ω ) . (cid:3) Theorem 4.14.
Consider an affine polynomial system ( f , . . . , f n ) in K [ x ] such that Assum. 4.1 holds and the system ( F , . . . , F n ) is(sparse) regular, where F i ∈ K [ S h ∆ ] e i and χ ( F i ) = f i , for i ∈ [ n ] .Then, the complexity of computing h f , . . . , f n i : h Î i x i i ∞ is O ( n + P ( d ) ω + n MV ( ∆ , . . . , ∆ n ) ) . Proof.
We need to compute: • The set
Rows ( ReduceMacaulay (( F , . . . , F n ) , Í i > e i , < )) to gen-erate L (Rem. 4.7). By Lem. 4.13, this costs O ( n + P ( Í i > e i ) ω ) . • The set
Rows ( ReduceMacaulay (( F , . . . , F n ) , , < )) to generatethe matrix M( F ) of Lem. 4.8. By Lem. 4.13, it costs O ( n + P ( ) ω ) . • For each variable x i , the Schur complement of M( F ) , for χ ( F ) = x i . The cost of each Schur complement computation is O ( P ( ) ω ) ,and so the cost of this step is O ( n P ( ) ω ) . • The complexity of FGLM depends on the number of solutions,and in this case it is O ( n MV ( ∆ , . . . , ∆ n ) ) [17]. (cid:3) Note that MV ( ∆ , . . . , ∆ n ) < P ( ) . Hence, to improve the previ-ous bound for lexicographical orders, we can follow [22]. Acknowledgements:
We thank C. D’Andrea, C. Massri, B. Mourrain, P.-J. Spaenle-hauer, and B. Teissier for the helpful discussions and references.
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A COUNTER-EXAMPLE TO THECOMPLEXITY BOUNDS IN [20]
Let ∆ be the standard 2-simplex and consider the regular systemgiven by two polynomials F , F ∈ K [ S h ∆ ] of degree , F : = X (( , ) , ) + X (( , ) , ) + X (( , ) , ) + X (( , ) , ) + X (( , ) , ) + X (( , ) , ) F : = X (( , ) , ) + X (( , ) , ) + X (( , ) , ) + X (( , ) , ) + X (( , ) , ) + X (( , ) , ) Consider the graded monomial order < given by X (( x , y ) , d ) < X (( x , y ) , d ) ⇐⇒ d < d , or d = d and x < x , or d = d and x = x and y < y . In this case, the bound in [20, Lem. 5.2] is , meanwhile the max-imal degree of an element in the Gröbner basis of ( F , F ) withrespect to < has degree . B ALGORITHM TO COMPUTE GRÖBNERBASIS OVER THE STANDARD ALGEBRA
Algorithm 3 compute-0-Dim-GB
Input:
Affine system f , . . . , f n ∈ K [ x ] and a monomial order < for K [ x ] , such that it has a finite number of solutions over ( C ∗ ) n and satisfies Assum. 4.1. Output: