Related papers: Hyperbolic polynomials and rigid moduli orders

We prove that if a polynomial has a root mod $p$ for every large prime $p$, then it has a real root. As an application, we show that the primes can't be covered by finitely many positive definite binary quadratic forms.

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 study sets of univariate hyperbolic polynomials that share the same first few coefficients and show that they have a natural combinatorial description akin to that of polytopes. We define a stratification of such sets in terms of root…

We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems---a class of convex optimization problems that generalize semidefinite programs. We show how to…

For a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$), Descartes' rule of signs says that $P$ has $pos\leq c$ positive and…

Let $p(x_1,...,x_n) =\sum_{(r_1,...,r_n) \in I_{n,n}} a_{(r_1,...,r_n)} \prod_{1 \leq i \leq n} x_{i}^{r_{i}}$ be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients. The support of such…

A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of strongly stable linear matrix…

The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not $C^{1,\alpha}$ for any $\alpha > 0$. A sufficient condition for the existence of a smooth parameterization is that no two of the…

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form…

We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence associated both to the…

Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z)$, $Z \in C^{n}$. We call such a polynomial $p$ {\bf H-Stable} if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1 \leq i \leq n$. This…

Given a polynomial $x \in {\mathbb R}^n \mapsto p(x)$ in $n=2$ variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set ${\mathcal P} = \{x : p(x) \geq 0\}$…

We give a strongly polynomial time algorithm which determines whether or not a bivariate polynomial is real stable. As a corollary, this implies an algorithm for testing whether a given linear transformation on univariate polynomials…

A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree…

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface…

In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus.…

The Hilbert class polynomial $H_{\mathcal{O}}(x)\in \mathbb{Z}[x]$ attached to an order $\mathcal{O}$ in an imaginary quadratic field $K$ is the monic polynomial whose roots are precisely the distinct $j$-invariants of elliptic curves over…

In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coeffcients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these…

We give a criterion which characterizes a homogeneous real multi-variate polynomial to have the property that all sufficiently large powers of the polynomial (as well as their products with any given positive homogeneous polynomial) have…

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to…