Related papers: Solving determinantal systems using homotopy techn…

Let $\mathbf{K}$ be a field of characteristic zero with $\overline{\mathbf{K}}$ its algebraic closure. Given a sequence of polynomials $\mathbf{g} = (g_1, \ldots, g_s) \in \mathbf{K}[x_1, \ldots , x_n]^s$ and a polynomial matrix $\mathbf{F}…

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 introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of rank $k$,…

By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure…

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of…

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A…

In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that…

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques…

Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution…

Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical methods. The polyhedral homotopy of Huber and Sturmfels for computing isolated solutions and the concept of witness sets as numerical…

For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed k>1, there is a polynomial-time algorithm that, for a 1-connected topological space X…

We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the…

In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list. We design an algorithm which solves systems with isolated solutions…

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but…

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…

Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula…

Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…

Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and…

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…

We present an algorithm that, given finite simplicial sets $X$, $A$, $Y$ with an action of a finite group $G$, computes the set $[X,Y]^A_G$ of homotopy classes of equivariant maps $\ell \colon X \to Y$ extending a given equivariant map $f…