相关论文: Toric Generalized Characteristic Polynomials

Sparse (or toric) elimination exploits the structure of polynomials by measuring their complexity in terms of Newton polytopes instead of total degree. The sparse, or Newton, resultant generalizes the classical homogeneous resultant and its…

This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the $u$-resultant and then retailor the generalized characteristic polynomial to fully exploit sparsity in the…

It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact…

Solving systems of polynomial equations is a central problem in nonlinear and computational algebra. Since Buchberger's algorithm for computing Gr\"obner bases in the 60s, there has been a lot of progress in this domain. Moreover, these…

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…

An improved characteristic set algorithm for solving Boolean polynomial systems is proposed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain…

Toric (or sparse) elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a \emph{semigroup algebra}, \emph{i.e.}…

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 theory of numerical path-following in toric varieties was suggested in two previous papers. The motivation is solving systems of polynomials with real or complex coefficients. When those polynomials are not assumed 'dense', solving them…

We study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one…

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…

We present our public-domain software for the following tasks in sparse (or toric) elimination theory, given a well-constrained polynomial system. First, C code for computing the mixed volume of the system. Second, Maple code for defining…

Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations, or sparse systems of exponential sums. The cost of continuation depends on a renormalized condition length,…

We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which in practice might…

Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…

In this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed…

We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the…

Finding the solutions to a system of multivariate polynomial equations is a fundamental problem in mathematics and computer science. It involves evaluating the polynomials at many points, often chosen from a grid. In most current methods,…

Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e. solving minimal…

We present a new method for solving symbolically zero--dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and…