Related papers: A criterion for the half-plane property
This paper extends characterizations of Sobolev spaces by Bourgain, Br\'{e}zis, and Mironescu to the higher order case. As a byproduct, we obtain an integral condition for the Taylor remainder term, which implies that the function is a…
We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup…
This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.
We investigate the strong Rayleigh property of matroids for which the basis enumerating polynomial is invariant under a Young subgroup of the symmetric group on the ground set. In general, the Grace-Walsh-Szeg\H{o} theorem can be used to…
Artin solved Hilbert's 17th problem, proving that a real polynomial in $n$ variables that is positive semidefinite is a sum of squares of rational functions, and Pfister showed that only $2^n$ squares are needed. In this paper, we…
There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also…
We introduce the concept of strong persistence and show that it implies persistence regarding the associated prime ideals of the powers of an ideal. We also show that strong persistence is equivalent to a condition on power of ideals…
The monostatic property of polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in a focus of research ever since Conway and Guy \cite{Conway} published the proof of the existence of the…
Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.
We use a variation on Mason's $\alpha$-function as a pre-dimension function to construct a not one-based $\omega$-stable plane $P$ (i.e. a simple rank $3$ matroid) which does not admit an algebraic representation (in the sense of matroid…
Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi--orthogonality conditions. We obtain several characterizations for these…
We show that if the ground set of a matroid can be partitioned into $k\ge 2$ bases, then for any given subset $S$ of the ground set, there is a partition into $k$ bases such that the sizes of the intersections of the bases with $S$ may…
We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but…
Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary parts of its roots are not contained in the relative interior of $K$. If $K$ is the…
Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach:…
The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions…
We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…
In this paper we provide some stability criteria for systems of linear subspaces of $V \otimes W$ and for systems of quotient coherent sheaves, using, respectively, the Hilbert-Mumford numerical criterion and moment map. Along the way, we…
The Rado-Horn theorem provides necessary and sufficient conditions for when a collection of vectors can be partitioned into a fixed number of linearly independent sets. Such partitions exist if and only if every subset of the vectors…
In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\Omega(n^{4/3})$, for every pair of…