Related papers: Hyperbolic polynomials and rigid moduli orders

In this paper we study the relationship between the set of all non-negative multivariate homogeneous polynomials and those, which we call hyperwrons, whose non-negativity can be deduced from an identity involving the Wronskians of…

Let R be a ring and let B be a commutative ring. Let p be a homogeneous multiplicative polynomial law of degree n from R to B. We show that p is essentially a determinant, in the sense that p is obtained from a determinant by left and right…

It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…

Poincare polinomials of hyperbolic Lie algebras, which are given by $HA_2$ and $HA_3$ in the Kac's notation, are calculated explicitly. The results show that there is a significant form for hyperbolic Poincare polinomials. Their explicit…

We study the class of polynomials whose Hessians evaluated at any point of a closed convex cone have Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. We prove that hyperbolic…

Given a real univariate degree $d$ polynomial $P$, the numbers $pos_k$ and $neg_k$ of positive and negative roots of $P^{(k)}$, $k=0$, $\ldots$, $d-1$, must be admissible, i.e. they must satisfy certain inequalities resulting from Rolle's…

Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic…

We consider the set of monic degree $d$ real univariate polynomials $Q_d=x^d+\sum_{j=0}^{d-1}a_jx^j$ and its {\em hyperbolicity domain} $\Pi_d$, i.e. the subset of values of the coefficients $a_j$ for which the polynomial $Q_d$ has all…

Given $H\subseteq \mathbb{C}$ two natural objects to study are the set of zeros of polynomials with coefficients in $H$, $$\{z\in \mathbb{C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum_{n=0}^{k}a_{n}z^n=0\},$$ and the set of zeros of…

We call a smooth function of one variable a degree n pseudopolynomial if its n-th derivative has no (real) zeros. An n pseudopolynomial is called hyperbolic if it has exactly n simple zeros. In this short note we describe the necessary and…

We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian…

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…

We consider complex Henon maps which are quasi-hyperbolic. We show that a quasi-hyperbolic map is uniformly hyperbolic if and only if there are no tangencies between stable and unstable manifolds.

We show that a monic univariate polynomial over a field of characteristic zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper leading coefficients by providing an explicit algorithm for computing the multiplicities…

We call a log variety (X, D) algebraically hyperbolic if there exists a positive number e such that 2g(C) - 2 + i(C, D) >= e deg(C) for all curves C on X, where i(C, D) is the number of the intersections between D and the normalization of…

In this paper, for positive integers $H$ and $k \leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include…

We show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation. This is proved by considering sum-of-squares decompositions of certain bilinear forms…

We prove that the roots of a definable $C^\infty$ curve of monic hyperbolic polynomials admit a definable $C^\infty$ parameterization, where `definable' refers to any fixed o-minimal structure on $(\mathbb R,+,\cdot)$. Moreover, we provide…

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…

Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex…