Related papers: Detecting rigid convexity of bivariate polynomials

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…

A real univariate polynomial of degree $n$ is called hyperbolic if all of its $n$ roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of…

A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for…

Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an…

This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…

We consider systems of strict multivariate polynomial inequalities over the reals. All polynomial coefficients are parameters ranging over the reals, where for each coefficient we prescribe its sign. We are interested in the existence of…

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…

This paper studies, for the first time, a bilevel polynomial program whose constraints involve uncertain linear constraints and another uncertain linear optimization problem. In the case of box data uncertainty, we present a sum of squares…

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…

Polynomial and rational functions are the number one choice when it comes to modeling of radial distortion of lenses. However, several extrapolation and numerical issues may arise while using these functions that have not been covered by…

This paper is about solving polynomial systems. It first recalls how to do that efficiently with a very high probability of correctness by reconstructing a rational univariate representation (rur) using Groebner revlex computation,…

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum…

For given integers $n$ and $d$, both at least 2, we consider a homogeneous multivariate polynomial $f_d$ of degree $d$ in variables indexed by the edges of the complete graph on $n$ vertices and coefficients depending on cardinalities of…

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…

We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has…

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…

Let T be the unit circle in the complex plane C. This paper proves the existence of analytic structure in a compact subset K of T X C^n, where K has so-called "lineally convex" or "hypoconvex" fibers over T. It also addresses a related…