arXiv subjects

Jérémy Berthomieu

Publications and source records attributed to Jérémy Berthomieu.

15 recordsLinked to original sources

arXiv2026

Computing submatrices of the Hermite normal form of a structured polynomial matrix

Following several decades of successive algorithmic improvements, works from the 2010s have showed how to compute the Hermite normal form (HNF) of a univariate polynomial matrix within a complexity bound which is essentially that of polynomial matrix multiplication. Recently, several results on bivariate polynomials and Gröbner bases have highlighted the interest of computing determinants or HNFs of polynomial matrices that happen to be structured, with a small displacement rank. In such contexts, a small leading principal submatrix of the HNF often contains all the sought information. In this article, we show how the displacement structure can be exploited in order to accelerate the computation of such submatrices. To achieve this, we rely on structured linear algebra over the field thanks to evaluation-interpolation. This allows us to recover some rows of the inverse of the input matrix, from which we deduce the sought HNF submatrix via bases of relations.

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arXiv2026

Computing points in connected components defined by a real inequation: algorithms, complexity and implementations, Part I

We consider the problem of computing sample points in each connected component of a semi-algebraic set defined by the non-vanishing or the positivity of an n-variate polynomial of degree d, with rational coefficients of bit size bounded by $τ$. Such a problem is a basic routine in effective real algebraic geometry, used in higher-level algorithms for solving polynomial systems over the reals and finds many applications in sciences. We design a probabilistic algorithm for solving this problem, which is based on reductions to different routines for solving zero-dimensional polynomial systems. It assumes that the input polynomial satisfies sufficiently generic properties (namely, smoothness of its defining hypersurface). This is done through the computations of critical points of well-chosen maps to capture the connected components of the semi-algebraic set under study. We derive a bit complexity estimate for the cost of this algorithm, which is, in terms of the B{é}zout bound d(d -1)^{n-1}, essentially cubic for obtaining parametrisations of the sought-for real points. Moreover, we also consider the case of obtaining rational approximations of those points, which are precise enough to lie in the same connected components as their exact counterparts, which yields a cost that is essentially quartic in the B{é}zout bound. In these complexity estimates, we take into account the degree structure of the input polynomial and its partial derivatives, allowing for a more refined bit complexity when the partial derivative of the input polynomial have degree lower than expected. We also analyse the probability of success of those algorithms. We report on practical experiments, benchmarking with random dense input polynomials as well as polynomials coming from applications, which were out of reach of the state-of-the-art implementations, and hence illustrate the practical efficiency of these new algorithms.

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arXiv2026

Multiword matrix multiplication over large finite fields in floating-point arithmetic

This article is concerned with the efficient computation of modular matrix multiplication C=AB mod p, a key kernel in computer algebra. We focus on floating-point arithmetic, which allows for using efficient matrix multiplication libraries. However, the existing approach is limited to primes p with bitsize at most half the mantissa size (e.g., 26 bits with double precision arithmetic), and becomes quite inefficient when p approaches this limit. We present a new approach that overcomes this limitation and can efficiently handle primes with larger bitsizes. The key idea is to use multiword decompositions, which represent A and B as scaled sums of u and v matrices (words) with smaller coefficients. We provide a rigorous analysis that proves the correctness of this approach for suitably chosen scaling parameters. Our analysis determines the maximum bitsize of p that can be handled for a given number of words; in particular, we show that decomposing in two words each input suffices to handle bitsizes almost equal to the full mantissa size (e.g., the 26 bits limit is raised to 52 bits in double precision arithmetic). Moreover, we show that (1,v) decompositions with v>1 are also of interest to handle intermediate bitsizes. We perform an extensive experimental analysis for various matrix shapes and prime bitsizes. Our performance benchmarks on both CPU and GPU architectures confirm the efficiency of the proposed approach, which can outperform the existing single word approach for bitsizes as low as 23, and can handle bitsizes as high as 52 while retaining high performance.

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arXiv2025

Extracting Linear Relations from Gröbner Bases for Formal Verification of And-Inverter Graphs

Formal verification techniques based on computer algebra have proven highly effective for circuit verification. The circuit, given as an and-inverter graph, is encoded as a set of polynomials that automatically generates a Gröbner basis with respect to a lexicographic term ordering. Correctness of the circuit can be derived by computing the polynomial remainder of the specification. However, the main obstacle is the monomial blow-up during the rewriting of the specification, which leads to the development of dedicated heuristics to overcome this issue. 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.

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arXiv2024

Computing Generic Fibers of Polynomial Ideals with FGLM and Hensel Lifting

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 $\mathfrak{m}$-adic expansions we compute. Some provided experimental data also demonstrates the practical efficacy of our algorithm.

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arXiv2023

New efficient algorithms for computing Gröbner bases of saturation ideals (F4SAT) and colon ideals (Sparse-FGLM-colon)

This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gröbner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical issue in non-linear algebra and more broadly in computational mathematics because of its numerous applications in engineering and computing sciences. Such applications often require geometric computing features such as representing the closure of the set difference of two solution sets to given polynomial systems. Algebraically, this boils down to computing Gröbner bases of colon and/or saturation polynomial ideals. In this paper, we describe and analyze new Gröbner bases algorithms for this task and present implementations which are more efficient by several orders of magnitude than the state-of-the-art software.

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arXiv2022

Faster change of order algorithm for Gröbner bases under shape and stability assumptions

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 \emph{shape} lexicographic Gröbner basis. Second, the set of leading terms with respect to the degree reverse lexicographic order has a \emph{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 $n \le t$, the arithmetic complexity of our algorithm is $O\tilde{~}(t^{ω-1}D)$, 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 $O\tilde{~}(tD^2)$ and $O\tilde{~}(D^ω)$, since $ω< 3$ and $t\le D$. Practical experiments, based on the libraries msolve and PML, confirm the high practical benefit.

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arXiv2022

Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems

We consider ideals involving the maximal minors of a polynomial matrix. For example, those arising in the computation of the critical values of a polynomial restricted to a variety for polynomial optimisation. Gröbner bases are a classical tool for solving polynomial systems. For practical computations, this consists of two stages. First, a Gröbner basis is computed with respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as \textsf{Sparse-FGLM}, designed by Faugère and Mou, is used to find a Gröbner basis of the same ideal but with respect to a lexicographic ordering. The complexity of this latter step, in terms of arithmetic operations, is $O(mD^2)$, where $D$ is the degree of the ideal and $m$ is the number of non-trivial columns of a certain $D \times D$ matrix. While asymptotic estimates are known for $m$ for generic polynomial systems, thus far, the complexity of \textsf{Sparse-FGLM} was unknown for determinantal systems. By assuming Fröberg's conjecture we expand the work of Moreno-Socías by detailing the structure of the DRL staircase in the determinantal setting. Then we study the asymptotics of the quantity $m$ by relating it to the coefficients of these Hilbert series. Consequently, we arrive at a new bound on the complexity of the \textsf{Sparse-FGLM} algorithm for generic determinantal systems and for generic critical point systems. We consider the ideal in the polynomial ring $\mathbb{K}[x_1, \dots, x_n]$, where $\mathbb{K}$ is some infinite field, generated by $p$ generic polynomials of degree $d$ and the maximal minors of a $p \times (n-1)$ polynomial matrix with generic entries of degree $d-1$. Then for the case $d=2$ and for $n \gg p$ we give an exact formula for $m$ in terms of $n$ and $p$. Moreover, for $d \geq 3$, we give an asymptotic formula, as $n \to \infty$, for $m$ in terms of $n,p$ and $d$.

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arXiv2021

Guessing Gr{ö}bner Bases of Structured Ideals of Relations of Sequences

Assuming sufficiently many terms of a n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In many applications, the table terms come along with a structure: for instance, they may be zero outside of a cone, they may be built from a Gr{ö}bner basis of an ideal invariant under the action of a finite group. Thus, we show how to take advantage of this structure to both reduce the number of table queries and the number of operations in the base field to recover the ideal of relations of the table. In applications like in combinatorics, where all these zero terms make us guess many fake relations, this allows us to drastically reduce these wrong guesses. These algorithms have been implemented and, experimentally, they let us handle examples that we could not manage otherwise. Furthermore, we show which kind of cone and lattice structures are preserved by skew polynomial multiplication. This allows us to speed up the guessing of linear recurrence relations with polynomial coefficients by computing sparse Gr{ö}bner bases or Gr{ö}bner bases of an ideal invariant under the action of a finite group in a ring of skew polynomials.

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arXiv2021

Polynomial-Division-Based Algorithms for Computing Linear Recurrence Relations

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 Gr{ö}bner 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 Pad{é} approximants of this mirror polynomial.As an addition from the paper published at the ISSAC conferance, we give an adaptive variant of this algorithm taking into account the shape of the final Gr{ö}bner basis gradually as it is discovered. The main advantage of this algorithm is that its complexity in terms of operations and sequence queries only depends on the output Gr{ö}bner basis.All these algorithms have been implemented in Maple and we report on our comparisons.

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arXiv2021

msolve: A Library for Solving Polynomial Systems

We present a new open source C library \texttt{msolve} dedicated to solving multivariate polynomial systems of dimension zero through computer algebra methods. The core algorithmic framework of \texttt{msolve} relies on Gr\''obner bases and linear algebra based algorithms for polynomial system solving. It relies on Gr\''obner basis computation w.r.t.\ the degree reverse lexicographical order, Gr\''obner conversion to a lexicographical Gr\''obner 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 \texttt{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 \texttt{msolve} with leading computer algebra systems such as \textsc{Magma}, \textsc{Maple}, \textsc{Singular} on a wide range of systems with finitely many complex solutions, showing that \texttt{msolve} can tackle systems which were out of reach by the computer algebra software state-of-the-art.

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arXiv2021

On the computation of asymptotic critical values of polynomial maps and applications

Let $\mathbf{f} = \left(f_1, \dots, f_p\right) $ be a polynomial tuple in $\mathbb{Q}[z_1, \dots, z_n]$ and let $d = \max_{1 \leq i \leq p} °f_i$. We consider the problem of computing the set of asymptotic critical values of the polynomial mapping, with the assumption that this mapping is dominant, $\mathbf{f}: z \in \mathbb{K}^n \to (f\_1(z), \dots, f\_p(z)) \in \mathbb{K}^p$ where $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$. This is the set of values $c$ in the target space of $\mathbf{f}$ such that there exists a sequence of points $(\mathbf{x}_i)_{i\in \mathbb{N}}$ for which $\mathbf{f}(\mathbf{x}_i)$ tends to $c$ and $\|\mathbf{x}_i\| κ{\rm d} \mathbf{f}(\mathbf{x}_i))$ tends to $0$ when $i$ tends to infinity where ${\rm d} \mathbf{f}$ is the differential of $\mathbf{f}$ and $κ$ is a function measuring the distance of a linear operator to the set of singular linear operators from $\mathbb{K}^n$ to $\mathbb{K}^p$. Computing the union of the classical and asymptotic critical values allows one to put into practice generalisations of Ehresmann's fibration theorem. This leads to natural and efficient applications in polynomial optimisation and computational real algebraic geometry. Going back to previous works by Kurdyka, Orro and Simon, we design new algorithms to compute asymptotic critical values. Through randomisation, we introduce new geometric characterisations of asymptotic critical values. This allows us to dramatically reduce the complexity of computing such values to a cost that is essentially $O(d^{2n(p+1)})$ arithmetic operations in $\mathbb{Q}$. We also obtain tighter degree bounds on a hypersurface containing the asymptotic critical values, showing that the degree is at most $p^{n-p+1}(d-1)^{n-p}(d+1)^{p}$. Next, we show how to apply these algorithms to unconstrained polynomial optimisation problems and the problem of computing sample points per connected component of a semi-algebraic set defined by a single inequality/inequation. We report on the practical capabilities of our implementation of this algorithm. It shows how the practical efficiency surpasses the current state-of-the-art algorithms for computing asymptotic critical values by tackling examples that were previously out of reach.

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arXiv2018

In-depth comparison of the Berlekamp--Massey--Sakata and the Scalar-FGLM algorithms: the adaptive variants

The Berlekamp--Massey--Sakata algorithm and the Scalar-FGLM algorithm both compute the ideal of relations of a multidimensional linear recurrent sequence.Whenever quering a single sequence element is prohibitive, the bottleneck of these algorithms becomes the computation of all the needed sequence terms. As such, having adaptive variants of these algorithms, reducing the number of sequence queries, becomes mandatory.A native adaptive variant of the Scalar-FGLM algorithm was presented by its authors, the so-called Adaptive Scalar-FGLM algorithm.In this paper, our first contribution is to make the Berlekamp--Massey--Sakata algorithm more efficient by making it adaptive to avoid some useless relation test-ings. This variant allows us to divide by four in dimension 2 and by seven in dimension 3 the number of basic operations performed on some sequence family.Then, we compare the two adaptive algorithms. We show that their behaviors differ in a way that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other. We detail precisely the differences and the similarities of both algorithms and conclude that in general the Adaptive Scalar-FGLM algorithm needs fewer queries and performs fewer basic operations than the Adaptive Berlekamp--Massey--Sakata algorithm.We also show that these variants are always more efficient than the original algorithms.

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arXiv2017

In-depth comparison of the Berlekamp -- Massey -- Sakata and the Scalar-FGLM algorithms: the non adaptive variants

We compare thoroughly the Berlekamp -- Massey -- Sakata algorithm and the Scalar-FGLM algorithm, which compute both the ideal of relations of a multi-dimensional linear recurrent sequence. Suprisingly, their behaviors differ. We detail in which way they do and prove that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other.

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arXiv2015

Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case

Let $\mathbf{f}=(f\_1,\ldots,f\_m)$ and $\mathbf{g}=(g\_1,\ldots,g\_m)$ be two sets of $m\geq 1$ nonlinear polynomials over $\mathbb{K}[x\_1,\ldots,x\_n]$ ($\mathbb{K}$ being a field). We consider the computational problem of finding -- if any -- an invertible transformation on the variables mapping $\mathbf{f}$ to $\mathbf{g}$. The corresponding equivalence problem is known as {\tt Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental problem in multivariate cryptography. The main result is a randomized polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a particular case of importance in cryptography and somewhat justifying {\it a posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt IP1S} for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory, which involves to test the orthogonal simultaneous conjugacy of symmetric matrices. We show that we can essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding an invertible matrix in the linear space $\mathbb{K}^{n \times n}$ of $n \times n$ matrices over $\mathbb{K}$ and to compute the square root in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing the square root in $\mathbb{K}^{n \times n}$ for various fields (including finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt IP1S} for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general {\it Isomorphism of Polynomials} ({\tt IP}) problem where we allow an invertible linear transformation on the variables \emph{and} on the set of polynomials. A randomized polynomial-time algorithm for solving {\tt IP} when $\mathbf{f}=(x\_1^d,\ldots,x\_n^d)$ is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (\emph{i.e.}\ a matrix whose components are linear polynomials). This extends to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.

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