Related papers: Hyperbolicity cones and imaginary projections

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous…

In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta,…

We give an elementary proof for the fact that an irreducible hyperbolic polynomial has only one pair of hyperbolicity cones.

Hyperbolic polynomials have been of recent interest due to applications in a wide variety of fields. We seek to better understand these polynomials in the case when they are symmetric, i.e. invariant under all permutations of variables. We…

We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones is given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans…

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by…

The set of homogeneous polynomials of degree $D$ is a topological space that contains the subspace $Hyp(D)$ constituted only by hyperbolic polynomials. In 2002, V. I. Arnold conjectured that the number of connected components of $Hyp (D)$…

For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the…

A variety of codimension $c$ in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension $c$. Positively hyperbolic hypersurfaces are defined by…

Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In…

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…

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly…

We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of…

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…

In this paper we extend the notion of a Lorentz cone. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., monotone) with respect to the order…

In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In…

A family $f_t(z)$ of polynomials over a number field $K$ will be called \emph{weighted homogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\geq 2$. For example, the family $z^d+t$ is…

The starting point of this paper is the computation of minimal hyperbolic polynomials of duals of cones arising from chordal sparsity patterns. From that, we investigate the relation between ranks of homogeneous cones and their minimal…

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…

Let $f,g \in k[x]$ be nonconstant polynomials over a number field $k$. We count $S$-integer inputs $a$ for which $f(a)$ has a $k$-rational preimage under $g$, after removing the polynomial graph components $Y=h(X)$ with $f=g\circ h$. The…