Related papers: Towards Incremental Cylindrical Algebraic Decompos…
The Conflict-Driven Cylindrical Algebraic Covering algorithm has proven well suited for performing theory validation checks in the satisfiability modulo theories paradigm for non-linear real arithmetic. CDCAC repurposes the theory…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
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…
We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both…
The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric…
We examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points. We show how interpolation matrices can be used for ray shooting of a parametric ray with a…
Gr\"obner Bases and Cylindrical Algebraic Decomposition are generally thought of as two, rather different, methods of looking at systems of equations and, in the case of Cylindrical Algebraic Decomposition, inequalities. However, even for a…
We describe a Maple package that serves at least four purposes. First, one can use it to compute whether or not a given polyhedral structure is Zometool constructible. Second, one can use it to manipulate Zometool objects, for example to…
Mathematical reasoning is a hallmark of human intelligence, requiring logical deduction, symbolic manipulation, and abstract thinking. Recent multimodal large language models (MLLMs) have demonstrated strong performance on geometry problems…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
We investigate the distribution of cells by dimension in cylindrical algebraic decompositions (CADs). We find that they follow a standard distribution which seems largely independent of the underlying problem or CAD algorithm used. Rather,…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
Modern advances in general-purpose computer algebra systems offer solutions to a variety of problems, which in the past required substantial time investments by trained mathematicians. An excellent example of such development are the…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
We use isomorphism $\varphi$ between matrix algebras and simple orthogonal Clifford algebras $\cl(Q)$ to compute matrix exponential ${e}^{A}$ of a real, complex, and quaternionic matrix A. The isomorphic image $p=\varphi(A)$ in $\cl(Q),$…
Let $\mathbb{Q}$ (resp. $\mathbb{R}$) be the field of rational (resp. real) numbers and $X = (X_1, \ldots, X_n)$ be variables. Deciding the non-negativity of polynomials in $\mathbb{Q}[X]$ over $\mathbb{R}^n$ or over semi-algebraic domains…
Within the framework of the augmented Lagrangian (AL), we propose a novel distributed optimization method, termed Distributed Augmented Lagrangian Decomposition (DALD), and provide a rigorous convergence proof for its standard version. To…
Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this…
The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation…