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Computing recurrence relations for sequences is a central problem in computer algebra, with applications in error-correcting codes, Gröbner basis computation, and sparse interpolation. While uni-indexed C-recursive sequences benefit from quasi-linear algorithms leveraging the half-gcd method, the extension to multi-indexed sequences remains computationally challenging. Existing methods for bi-indexed sequences achieve quadratic complexity at best, limiting their practical use.

Abstract Formal verification techniques based on computer algebra have proven highly effective for circuit verification. The circuit, given as an and-inverter graph, is encoded using polynomials that automatically generate a Gröbner basis with respect to a lexicographic term ordering. Correctness of the circuit is derived by computing the polynomial remainder of the specification. However, the main obstacle is the monomial blow-up during the reduction, as the degree can increase. In this paper, we investigate an orthogonal approach and focus the computational effort on rewriting the Gröbner basis itself. Our goal is to ensure the basis contains linear polynomials that can be effectively used to rewrite the linearized specification. We first prove the soundness and completeness of this technique and then demonstrate its practical application. Our implementation of this method shows promising results on benchmarks related to multiplier verification.

We describe a version of the FGLM algorithm that can be used to compute generic fibers of positive-dimensional polynomial ideals. It combines the FGLM algorithm with a Hensel lifting strategy. In analogy with Hensel lifting, we show that this algorithm has a complexity quasi-linear in the number of terms of certain <?TeX $\mathfrak {m}$?> Math 1 -adic expansions we compute. Some provided experimental data also demonstrates the practical efficacy of our algorithm.

Solving zero-dimensional polynomial systems using Gröbner bases is usually done by, first, computing a Gröbner basis for the degree reverse lexicographic order, and next computing the lexicographic Gröbner basis with a change of order algorithm. Currently, the change of order now takes a significant part of the whole solving time for many generic instances. Like the fastest known change of order algorithms, this work focuses on the situation where the ideal defined by the system satisfies natural properties which can be recovered in generic coordinates. First, the ideal has a shape lexicographic Gröbner basis. Second, the set of leading terms with respect to the degree reverse lexicographic order has a stability property; in particular, the multiplication matrix can be read on the input Gröbner basis. The current fastest algorithms rely on the sparsity of this matrix. Actually, this sparsity is a consequence of an algebraic structure, which can be exploited to represent the matrix concisely as a univariate polynomial matrix. We show that the Hermite normal form of that matrix yields the sought lexicographic Gröbner basis, under assumptions which cover the shape position case. Under some mild assumption implying n≤t, the arithmetic complexity of our algorithm is O~(tω-1D), where n is the number of variables, t is a sparsity indicator of the aforementioned matrix, D is the degree of the zero-dimensional ideal under consideration, and ω is the exponent of matrix multiplication. This improves upon both state-of-the-art complexity bounds O~(tD2) and O~(Dω, since ω<3 and t≤D. Practical experiments, based on the libraries msolve and PML, confirm the high practical benefit.

We present a new open source C library msolve dedicated to solving multivariate polynomial systems of dimension zero through computer algebra methods. The core algorithmic framework of msolve relies on Gröbner bases and linear algebra based algorithms for polynomial system solving. It relies on Gröbner basis computation w.r.t. the degree reverse lexicographical order, Gröbner conversion to a lexicographical Gröbner basis and real solving of univariate polynomials. We explain in detail how these three main steps of the solving process are implemented, how we exploit AVX2 instruction processors and the more general implementation ideas we put into practice to better exploit the computational capabilities of this algorithmic framework. We compare the practical performances of msolve with leading computer algebra systems such as Magma, Maple, Singular on a wide range of systems with finitely many complex solutions, showing that msolve can tackle systems which were out of reach by the computer algebra software state-of-the-art.

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp--Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp--Massey--Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp--Massey--Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. Finally, we give a partial solution to the transformation of this algorithm into an adaptive one.

Given several n-dimensional sequences, we first present an algorithm for computing the Grobner basis of their module of linear recurrence relations.

Sakata generalized the Berlekamp--Massey algorithm to n dimensions in~1988. The Berlekamp--Massey--Sakata (BMS) algorithm can be used for finding a Grbner basis of a 0-dimensional ideal of relations verified by a table. We investigate this problem usingö linear algebra techniques, with motivations such as accelerating change of basis algorithms (FGLM) or improving their complexity.

In a previous article [1], an implementation of lazy p-adic integers with a multiplication of quasi-linear complexity, the so-called relaxed product, was presented. Given a ring R and an element p in R, we design a relaxed Hensel lifting for algebraic systems from R/ (p) to the p-adic completion Rp of R. Thus, any root of linear and algebraic regular systems can be lifted with a quasi-optimal complexity. We report our implementations in C++ within the computer algebra system Mathemagix and compare them with Newton operator. As an application, we solve linear systems over the integers and compare the running times with Linbox and IML.