Related papers: Van der Waerden/Schrijver-Valiant like Conjectures…

We prove that if $P(X) \in \mathbb{Z}[X]$ is an integer polynomial of degree $n$ and having $P(0) = 1$, then either $P(X)$ is a product of cyclotomic polynomials, or else at least one of the complex roots of $P$ belongs to the disk $|z|…

In this paper, we investigate $\mathbb T^d$-invariant Hilbert modules $\mathscr H$ over the polynomial ring $\mathbb C[z_1, \ldots, z_d]$ and their quotients, with primary emphasis on the classification of subnormal quotient modules of the…

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 ${\mathcal P} = \{x : p(x) \geq 0\}$…

Given a separable nonconstant polynomial $f(x)$ with integer coefficients, we consider the set $S$ consisting of the squarefree parts of all the rational values of $f(x)$, and study its behavior modulo primes. Fixing a prime $p$, we…

Let $V$ be a vector space over a finite field $k=\mathbb{F} _q$ of dimension $n$. For a polynomial $P:V\to k$ we define the bias of $P$ to be $$b_1(P)=\frac {|\sum _{v\in V}\psi (P(V))|}{q^n}$$ where $\psi :k\to \mathbb{C} ^\star$ is a…

Let $P=\{p_{1},\ld,p_{r}\}\subset\Q[n_{1},\ld,n_{m}]$ be a family of polynomials such that $p_{i}(\Z^{m})\sle\Z$, $i=1,\ld,r$. We say that the family $P$ has {\it PSZ property} if for any set $E\sle\Z$ with…

We present a novel method for checking the Hurwitz stability of a polytope of matrices. First we prove that the polytope matrix is stable if and only if two homogenous polynomials are positive on a simplex, then through a newly proposed…

In this paper we are concerned with the Bohnenblust--Hille type inequalities for certain polynomials of bounded degree but of very large number of variables. As the polynomials will be defined on groups, one can think about the problem as…

For a given decreasing positive real function $\psi$, let $\mathcal{A}_n(\psi)$ be the set of real numbers for which there are infinitely many integer polynomials $P$ of degree up to $n$ such that $\left\lvert P(x) \right\rvert \leq…

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…

Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros…

A multivariate polynomial $p(x)=p(x_1,...,x_n)$ is sos-convex if its Hessian $H(x)$ can be factored as $H(x)= M^T(x) M(x)$ with a possibly nonsquare polynomial matrix $M(x)$. It is easy to see that sos-convexity is a sufficient condition…

A Boolean function $f:V \to \{-1,1\}$ on the vertex set of a graph $G=(V,E)$ is locally $p$-stable if for every vertex $v$ the proportion of neighbours $w$ of $v$ with $f(v)=f(w)$ is exactly $p$. This notion was introduced by Gross and…

Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential…

We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ \Delta_{h_0}^{m+1}f(x)=0 \ \ \text{for all}…

Let $P$ be a bounded convex subset of $\mathbb R^n$ of positive volume. Denote the smallest degree of a polynomial $p(X_1,\dots,X_n)$ vanishing on $P\cap\mathbb Z^n$ by $r_P$ and denote the smallest number $u\geq0$ such that every function…

Consider a logharmonic polynomial; that is, a product of the form $p(z)\overline{q(z)}$, where $p$, $q$ are holomorphic polynomials. Assume $q$ is linear and denote by $n$ the degree of $p$. It was recently shown in arXiv:2302.04339…

A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of the…

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension $n$. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension…

The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime…