Related papers: A Bivariate Polynomial Problem for Matrices
We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the…
Let $f, f_1, \ldots, f_\nV$ be polynomials with rational coefficients in the indeterminates $\bfX=X_1, \ldots, X_n$ of maximum degree $D$ and $V$ be the set of common complex solutions of $\F=(f_1,\ldots, f_\nV)$. We give an algorithm…
Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to…
The two-matrix model can be solved by introducing bi-orthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of bi-orthogonal polynomials (called "windows") satisfy polynomial ODEs as well as…
The equivalence of multidimensional systems is closely related to the reduction of multivariate polynomial matrices, with the Smith normal form of matrices playing a key role. So far, the problem of reducing multivariate polynomial matrices…
A vertex of a plane digraph is bimodal if all its incoming edges (and hence all its outgoing edges) are consecutive in the cyclic order around it. A plane digraph is bimodal if all its vertices are bimodal. Bimodality is at the heart of…
Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…
A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented. The key idea is to establish a relationship between a matrix and its full…
In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…
In this paper we present methods for the synthesis of polynomial invariants for probabilistic transition systems. Our approach is based on martingale theory. We construct invariants in the form of polynomials over program variables, which…
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level…
The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various…
We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm BISOLVE. BISOLVE is a classical elimination method which first projects the solutions of a system onto the $x$-…
A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue…
The invariant polytope algorithm was a breakthrough in the joint spectral radius computation, allowing to find the exact value of the joint spectral radius for most matrix families~\cite{GP2013,GP2016}. This algorithm found many…
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…
Given a set of point correspondences in two images, the existence of a fundamental matrix is a necessary condition for the points to be the images of a 3-dimensional scene imaged with two pinhole cameras. If the camera calibration is known…
The row (column) completion problem of polynomial matrices of given degree with prescribed eigenstructure has been studied in \cite{AmBaMaRo23}, where several results of prescription of some of the four types of invariants that form the…
In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard…
In this paper, we study a class of boundary value problems (BVPs) with Robin conditions in some $L^p$ spaces for polyharmonic equation on Lipschitz domains. Utilizing polyharmonic fundamental solutions, these Robin BVPs are solved by the…