Related papers: Generic matrix polynomials with fixed rank and fix…

We show that a real homogeneous polynomial f(x,y) with distinct roots and degree d greater or equal than 3 has d real roots if and only if for any (a,b) not equal to (0,0) the polynomial af_x+bf_y has d-1 real roots. This answers to a…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

Let P be an elementary closed semi-algebraic set in R^d, i.e., there exist real polynomials p_1,...,p_s such that P= \{x \in R^d : p_1(x) \ge 0, >..., p_s(x) \ge 0 \}; in this case p_1,...,p_s are said to represent P. Denote by $n$ the…

We introduce a new concept of rank - relative rank associated to a filtered collection of polynomials. When the filtration is trivial our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of…

Under the action of the general linear group with tensor structure, the ranks of matrices $A$ and $B$ forming an $m \times n$ pencil $A + \lambda B$ can change, but in a restricted manner. Specifically, with every pencil one can associate a…

Here we study the typical rank for real bivariate homogeneous polynomials of degree $d\ge 6$ (the case $d\le 5$ being settled by P. Comon and G. Ottaviani). We prove that $d-1$ is a typical rank and that if $d$ is odd, then $(d+3)/2$ is a…

Our results concern minimal graded free resolutions of fat point ideals for points in a hyperplane. Suppose, for example, that I(m,d) is the ideal defining r given points of multiplicity m in the projective space P^d. Assume that the given…

We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$…

A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of…

We solve the problem of characterizing the existence of a polynomial matrix of fixed degree when its eigenstructure (or part of it) and some of its rows (columns) are prescribed. More specifically, we present a solution to the row (column)…

This article introduces a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed rectangular region. We call this the branched…

The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the…

We characterize the existence of a polynomial (rational) matrix when its eigenstructure (complete structural data) and some of its rows are prescribed. For polynomial matrices, this problem was solved in a previous work when the polynomial…

Let Pd,n denote the space of all real polynomials of degree at most d on R^n. We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P in Pd,1. Using this estimate, we prove a sharp estimate for a singular…

This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at…

For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\cdots +a_n$, $n\geq 4$, we consider its higher-order discriminant sets $\{ \tilde{D}_m=0\}$, where $\tilde{D}_m:=$Res$(P,P^{(m)})$, $m=2$, $\ldots$, $n-2$, and their…

Let $P_1,...,P_n$ be generic homogeneous polynomials in $n$ variables of degrees $d_1,...,d_n$ respectively. We prove that if $\nu$ is an integer satisfying ${\sum_{i=1}^n d_i}-n+1-\min\{d_i\}<\nu,$ then all multivariate subresultants…

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…

We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…

Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with…