Computing syzygies in finite dimension using fast linear algebra Vincent Neiger Univ. Limoges, CNRS, XLIM, UMR 7252, F-87000 Limoges, France Eric´ Schost University of Waterloo, Waterloo ON, Canada Abstract We consider the computation of syzygies of multivariate polynomials in a finite-dimensional setting: for a K[X1;:::; Xr]-module M of finite dimension D as a K-vector space, and given elements f1;:::; fm in M, the problem is to compute syzygies between the fi’s, that is, poly- m nomials (p1;:::; pm) in K[X1;:::; Xr] such that p1 f1 + ··· + pm fm = 0 in M. Assuming that the multiplication matrices of the r variables with respect to some basis of M are known, we give an algorithm which computes the reduced Grobner¨ basis of the module of these syzygies, − for any monomial order, using O(mD! 1 + rD! log(D)) operations in the base field K, where ! is the exponent of matrix multiplication. Furthermore, assuming that M is itself given as n M = K[X1;:::; Xr] =N, under some assumptions on N we show that these multiplication matri- ces can be computed from a Grobner¨ basis of N within the same complexity bound. In particular, taking n = 1, m = 1 and f1 = 1 in M, this yields a change of monomial order algorithm along the lines of the FGLM algorithm with a complexity bound which is sub-cubic in D. Keywords: Grobner¨ basis, syzygies, complexity, fast linear algebra. 1. Introduction In what follows, K is a field and we consider the polynomial ring K[X] = K[X1;:::; Xr]. The set of m × n matrices over a ring R is denoted by Rm×n; when orientation matters, a vector in Rn is considered as being in R1×n (row vector) or in Rn×1 (column vector). We are interested in the efficient computation of relations, known as syzygies, between elements of a K[X]-module M. Let us write the K[X]-action on M as (p; f ) 2 K[X] × M 7! p · f , and let f1;:::; fm be in M. Then, for a given monomial order ≺ on K[X]m, we want to compute the Grobner¨ basis of the kernel of the homomorphism arXiv:1912.01848v2 [cs.SC] 19 Jun 2020 K[X]m !M (p1;:::; pm) 7! p1 · f1 + ··· + pm · fm: This kernel is called the module of syzygies of ( f1;:::; fm) and written SyzM( f1;:::; fm). In this paper, we focus on the case where M has finite dimension D as a K-vector space; as a m result, the quotient K[X] = SyzM( f1;:::; fm) has dimension at most D as a K-vector space. Then one may adopt a linear algebra viewpoint detailed in the next paragraph, where the elements of M are seen as row vectors of length D and the multiplication by the variables is represented by so-called multiplication matrices. This representation was used and studied in [2, 37,1, 27], mainly in the context where M is a quotient K[X]=I for some ideal I (thus zero-dimensional of degree D) and more generally a quotient K[X]n=N for some submodule N ⊆ K[X]n with n 2 N>0 (see [1, Sec. 4.4 and 6]). This representation with multiplication matrices allows one to perform computations in such a quotient via linear algebra operations. Assume we are given a K-vector space basis F of M. For i in f1;:::; rg, the matrix of the structure morphism f 7! Xi · f with respect to this basis is denoted by Mi; this means that for f in 1×D M represented by the vector f 2 K of its coefficients on F , the coefficients of Xi · f 2 M on F are f Mi. We call M1;:::; Mr multiplication matrices; note that they are pairwise commuting. The data formed by these matrices defines the module M up to isomorphism; we use it as a × representation of M. For p in K[X] and for f in M represented by the vector f 2 K1 D of its coefficients on F , the coefficients of p · f 2 M on F are f p(M1;:::; Mr); hereafter this vector is written p · f. From this point of view, syzygy modules can be described as follows. Definition 1.1. For m and D in N>0, let M = (M1;:::; Mr) be pairwise commuting matrices in D×D m×D m K , and let F 2 K . Denoting by f1;:::; fm the rows of F, for p = (p1;:::; pm) 2 K[X] we write 1×D p · F = p1 · f1 + ··· + pm · fm = f1 p1(M) + ··· + fm pm(M) 2 K : The syzygy module SyzM(F), whose elements are called syzygies for F, is defined as m SyzM(F) = fp 2 K[X] j p · F = 0g; m as noted above, K[X] = SyzM(F) has dimension at most D as a K-vector space. In particular, if in the above context F is the matrix of the coefficients of f1;:::; fm 2 M on the basis F , then SyzM(F) = SyzM( f1;:::; fm). Our main goal in this paper is to give a fast algorithm to solve the following problem (for the notions of monomial orders and Grobner¨ basis for modules, we refer to [13] and the overview in Section2). Problem 1 – Gr¨obnerbasis of syzygies Input: • a monomial order ≺ on K[X]m, D×D • pairwise commuting matrices M = (M1;:::; Mr) in K , × • a matrix F 2 Km D. Output: the reduced ≺-Grobner¨ basis of SyzM(F). Example 1.2. An important class of examples has r = 1; in this case, we are working with D univariate polynomials. Restricting further, consider the case M = K[X1]=hX1 i endowed with the canonical K[X1]-module structure; then M1 is the upper shift matrix, whose entries are all 0 m except those on the superdiagonal which are 1. Given f1;:::; fm in M,(p1;:::; pm) 2 K[X1] is a D syzygy for f1;:::; fm if p1 f1 +···+pm fm = 0 mod X1 . Such syzygies are known as Hermite-Pad´e D approximants of ( f1;:::; fm)[22, 40]. Using moduli other than X1 leads one to generalizations such as M-Pad´eapproximants or rational interpolants (corresponding to a modulus that splits into linear factors) [33,4, 49]. For r = 1, SyzM(F) is known to be free of rank m. Bases of such K[X1]-modules are often described by means of their so-called Popov form [43, 26]. In commutative algebra terms, this 2 is a term over position Grobner¨ basis. Another common choice is the Hermite form, which is a position over term Grobner¨ basis [29]. Example 1.3. For arbitrary r, let I be a zero-dimensional ideal in K[X] and let M = K[X]=I with the canonical K[X]-module structure. Then, taking m = 1 and f1 = 1 2 M, we have SyzM( f1) = fp 2 K[X] j p f1 = 0g = fp 2 K[X] j p 2 Ig = I: Suppose we know a Grobner¨ basis of I for some monomial order ≺1, together with the cor- responding monomial basis of M, and the multiplication matrices of X1;:::; Xr in M. Then solving Problem1 amounts to computing the Gr obner¨ basis of I for the new order ≺. More generally for a given f1 = f 2 M, the case m = 1 corresponds to the computation of f g the annihilator of f in K[X], often denoted by AnnK[X]( f ). Indeed, the latter set is defined as fp 2 K[X] j p f = 0g, which is precisely SyzM( f ). r Example 1.4. For an arbitrary r, let α1;:::; αD be pairwise distinct points in K , with αi = (αi;1; : : : ; αi;r) for all i. Let I be the vanishing ideal of fα1;:::; αDg and M = K[X]=I. As above, take m = 1 and f1 = 1, so that SyzM(1) = I. The Chinese Remainder Theorem gives an explicit isomorphism M! KD that amounts to D evaluation at α1;:::; αD. The multiplication matrices induced by this module structure on K are 1×D diagonal, with M j having diagonal (α1; j; : : : ; αD; j) for 1 6 j 6 r. Taking F = [1 ··· 1] 2 K , solving Problem1 allows us to compute the Gr obner¨ basis of the vanishing ideal I for any given order ≺. This problem was introduced by Moller¨ and Buchberger [36]; it may be extended to cases where vanishing multiplicities are prescribed [35]. Example 1.5. Now we consider an extension of the Moller-Buchberger¨ problem due to Kehrein, Kreuzer and Robbiano [27]. Given r pairwise commuting d × d matrices N1;:::; Nr, we look for their ideal of syzygies, that is, the ideal I of polynomials p 2 K[X] such that p(N1;:::; Nr) = 0. When r = 1, this ideal is generated by the minimal polynomial of N1. × One may see this problem in our framework by considering M = Kd d endowed with the d×d K[X]-module structure given by Xk · A = ANk for all 1 6 k 6 r and A 2 K . The ideal I defined above is the module of syzygies SyzM( f ) of the identity matrix f = Id 2 M, so we have m = d and D = d2 here. To form the input of Problem1, we choose as a basis of M the list of d×d elementary matrices F = (c1;1;:::; c1;d;:::; cd;1;:::; cd;d) where ci; j is the matrix in K whose only nonzero entry is a 1 at index (i; j).