Related papers: A stratified polyhedral homotopy method for sampli…
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…
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…
In this paper, we compute the homology group and cohomology algebra of various polyhedral product objects uniformly from the point of view of diagonal tensor product. As applications, we introduce the polyhedral product method into…
Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by…
Numerical algebraic geometry provides a number of efficient tools for approximating the solutions of polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method to solve numerous polynomial…
We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety $X_\Sigma$. The algorithm lends its name from a construction, described by Cox, of $X_\Sigma$ as a GIT quotient $X_\Sigma…
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…
Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
A polynomial homotopy is a family of polynomial systems in one parameter, which defines solution paths starting from known solutions and ending at solutions of a system that has to be solved. We consider paths leading to isolated singular…
Inspired by numerical homotopy methods we propose a combinatorial homotopy algorithm for finding all isolated solutions to a tropical polynomial systems of n tropical polynomials in n variables. In particular, a tropicalisation of the…
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…
A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can…
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}…
A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial…
This paper introduces an algebraic combinatorial approach to simplicial cone decompositions, a key step in solving inhomogeneous linear Diophantine systems and counting lattice points in polytopes. We use constant term manipulation on the…
We introduce a simple calculus, extending a variant of the Steenbrink spectrum, for describing Hodge-theoretic invariants of (smoothings of) isolated singularities with (relative) automorphisms. After computing these "eigenspectra" in the…
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a…
By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper…