Symbolic Computation
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…
When a computer algebra system fails to solve an Ordinary Differential Equation, is this a limitation of its implementation, or a genuine computational barrier? Three traditions bear on the question. Modern computer algebra algorithms can…
We introduce Exhaustive Symbolic Integration (ESI), a method that enumerates all symbolic functions up to a given complexity $k$ within a specified operator basis and determines which admit closed-form antiderivatives within the same class.…
Cylindrical Algebraic Decomposition (CAD) algorithms typically produce a decomposition adapted to a finite family of semi-algebraic sets $\mathcal{F}$ (i.e. every member of $\mathcal{F}$ is a union of cells). Different algorithms may…
Word-level verification of arithmetic circuits with large operands typically relies on arbitrary-precision arithmetic, which can lead to significant computational overhead as word sizes grow. In this paper, we present a hybrid algebraic…
This paper focuses on asymptotic properties of random monomial ideals through a statistical viewpoint. It extends the study of redundancy in monomial ideals by analyzing the poset density of the LCM-lattice. We explore how this density…
The Weyl closure is a basic operation in algebraic analysis: it converts a system of differential operators with rational coefficients into an equivalent system with polynomial coefficients. In addition to encoding finer information on the…
We give a self-contained, modern exposition of \'Edouard Goursat's 1887 theorem on pseudo-elliptic integrals -- those integrals of the form $\int F(t)\,\d t/\sqrt{R(t)}$ with $R$ a cubic or quartic polynomial that, despite living on a…
We describe the design of a quantifier elimination framework for the complex numbers in the language of ordered rings supplemented with symbols for the imaginary unit, real parts, imaginary parts, and conjugates. Technically, we use a…
We design the Arboretum$.$hs package for symbolic computations with algebras of trees and more general graphs in Haskell. Thanks to the declarative nature of functional programming, the package's implementation closely follows mathematical…