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Mathematical reasoning is an important research direction in the field of artificial intelligence. This article proposes a novel multi tool application framework for mathematical reasoning, aiming to achieve more comprehensive and accurate…
We present {\tt RandomPoints}, a package in \emph{Macaulay2} designed mainly to identify rational and geometric points in a variety over a finite field. We provide tools to estimate the dimension of a variety. We also present methods to…
The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections…
We consider systems of polynomial equations and inequalities in $\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1, \ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$ indeterminates are…
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…
Large language models (LLMs) excel at solving complex tasks by executing agentic workflows composed of detailed instructions and structured operations. Yet, building general-purpose agents by manually embedding foundation models into…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
The goal of this paper is to survey the properties of the eigenvalue relaxation for least squares binary problems. This relaxation is a convex program which is obtained as the Lagrangian dual of the original problem with an implicit compact…
This document describes our freely distributed Maple library {\sc spectra}, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities with symbolic computation in exact…
POCP is a new Matlab package running jointly with GloptiPoly 3 and, optionally, YALMIP. It is aimed at nonlinear optimal control problems for which all the problem data are polynomial, and provides an approximation of the optimal value as…
Specialized function gradient computing hardware could greatly improve the performance of state-of-the-art optimization algorithms, e.g., based on gradient descent or conjugate gradient methods that are at the core of control, machine…
This paper proposes a novel and simple algorithm of facet enumeration for convex polytopes. The complexity of the algorithm is discussed. The algorithm is implemented in Matlab. Some simple polytopes with known H-representations and…
Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the…
Multi-label learning has attracted the attention of the machine learning community. The problem conversion method Binary Relevance converts a familiar single label into a multi-label algorithm. The binary relevance method is widely used…
Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications within algebraic geometry and beyond. We recently reported on a new implementation of CAD in Maple which…
Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlev\'e test, which is applicable to…
In this article, we describe an implementation of a polynomial system solver to compute the approximate solutions of a 0-dimensional polynomial system with finite precision p-adic arithmetic. We also describe an improvement to an algorithm…
A generalized eigenvalue algorithm for tridiagonal matrix pencils is presented. The algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some…
We introduce efficient differentially private (DP) algorithms for several linear algebraic tasks, including solving linear equalities over arbitrary fields, linear inequalities over the reals, and computing affine spans and convex hulls. As…
Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…