符号计算
Gr\"obner basis computation incurs heavy computational overhead, especially under lexicographic order. F5 and its GVW variant dominate efficient field-based Gr\"obner basis solving. The proper basis algorithm offers a parameterized ideal…
In this note, we present RationalUnivariateRepresentation.jl (https://newrur.gitlabpages.inria.fr/RationalUnivariateRepresentation.jl/), a Julia package for computing rational univariate representations of zero-dimensional polynomial…
A standard way to control expression swell in computer algebra is to use multi-modular or evaluation-interpolation methods. In computations involving Gr\"obner bases, these techniques typically require repeatedly computing Gr\"obner bases…
Let $p$ be a prime and $R=\mathbb{Z}/p^2\mathbb{Z}$ the ring of integers modulo $p^2$. Any $A\in R^{n\times n}$ is unimodularly equivalent to its Smith form \[ S=diag\bigl(\underbrace{1,\ldots,1}_{r_0}, \underbrace{p,\ldots,p}_{r_1},…
We present a new algebraic modeling of the Supersingular Isogeny Problem as a system of multivariate polynomial equations, in the case where the elliptic curves are connected by an isogeny whose degree is a power of $2$ or $3$. This…
We improve significantly the Nart-Montes algorithm for factoring polynomials over a complete discrete valuation ring $\mathbb{A}$. Our first contribution is to extend the Hensel lemma in the context of generalised Newton polygons, from…
We present an experimental evaluation of automatically generated polynomial symmetry breaking constraints for integer linear programs. Starting from the method that we introduced at the International Symposium on Symbolic and Algebraic…
In this note, we study the complexity of multiplication in skew polynomial rings over finite fields. We prove that the product of two elements in $\mathbb{F}_{q^n}[x;\sigma]$ of degree at most $d < n$ can be computed using $\widetilde…
Let $\mathsf{E}=\mathbb F_q[x]/(\Gamma)$ be an algebraic extension of degree $n$ over the finite field $\mathbb F_q$, given by a $\Gamma\in\mathbb F_q[x]$ monic and irreducible. It is classical that any such $\mathsf{E}$ contains an element…
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…
Large Language Models (LLMs) have demonstrated impressive progress in complex reasoning tasks, largely driven by the Chain-of-Thought (CoT) paradigm, which decomposes difficult problems into intermediate steps. However, CoT reasoning…
We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…
We present a new algorithm for fast matrix multiplication using tensor decompositions which have special features. Thanks to these features we obtain exponents lower than what the rank of the tensor decomposition suggests. In particular for…
This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of…
A new symbolic algorithm to compute sums of squares multipliers (certificates) to witness the membership of non-negative univariate polynomials in a saturated univariate quadratic module is presented. Certificates are first computed in…
Objective: Acute mountain sickness (AMS) is the most prevalent altitude illness, affecting unacclimatized individuals ascending above 2,500 m and potentially escalating to life threatening cerebral or pulmonary edema. Conventional machine…
The positivity of the Gram-Charlier probability density function has been a subject of extensive study for decades. Since Barton and Dennis (1952) introduced numerical positivity conditions, no analytic closed-form expression was available…
A fundamental challenge in symbolic regression (SR) is efficiently recovering complex mathematical expressions from observational data. Although this problem is NP-hard, many expressions of practical interest decompose naturally into…
We introduce monomial divisibility diagrams (MDDs), a data structure for monomial ideals that supports insertion of new generators and fast membership tests. MDDs stem from a canonical tree representation by maximally sharing equal…
We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties…