Related papers: Polynomials vanishing at lattice points in a conve…

Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed $k\geq 1$ and $n$ large with respect to $k$, what is the minimum possible…

A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of…

We study the following natural question on random sets of points in $\mathbb{F}_2^m$: Given a random set of $k$ points $Z=\{z_1, z_2, \dots, z_k\} \subseteq \mathbb{F}_2^m$, what is the dimension of the space of degree at most $r$…

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev $p$-norm ($1<p<\infty$) for the case $p=1$. Some relevant examples are indicated. The second part deals…

We show that a $d$-dimensional polyhedron $S$ in $\real^d$ can be represented by $d$-polynomial inequalities, that is, $S = \{x \in \real^d : p_0(x) \ge 0, >..., p_{d-1}(x) \ge 0 \}$, where $p_0,...,p_{d-1}$ are appropriate polynomials.…

Let P be an elementary closed semi-algebraic set in R^d, i.e., there exist real polynomials p_1,...,p_s such that P= \{x \in R^d : p_1(x) \ge 0, >..., p_s(x) \ge 0 \}; in this case p_1,...,p_s are said to represent P. Denote by $n$ the…

For an integral convex polytope $\mathcal{P} \subset \mathbb{R}^d$, we recall $L_\mathcal{P}(n)=|n\mathcal{P} \cap \mathbb{Z}^d|$ the Ehrhart polynomial of $\mathcal{P}$. Let $g_r(\mathcal{P})$ be the $r$th coefficients of…

For a convex lattice polytope $P\subset \mathbb R^d$ of dimension $d$ with vertices in $\mathbb Z^d$, denote by $L(P)$ its discrete volume which is defined as the number of integer points inside $P$. The classical result due to Ehrhart says…

Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We…

Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$.…

We study residual polynomials, $R_{x_0,n}^{(\mathfrak{e})}$, $\mathfrak{e}\subset\mathbb{R}$, $x_0\in\mathbb{R}\setminus\mathfrak{e}$, which are the degree at most $n$ polynomials with $R(x_0)=1$ that minimize the $\sup$ norm on…

In this article we prove that the adjoint polynomial of arbitrary convex polytopes is up to scaling uniquely determined by vanishing to the right order on the polytopes residual arrangement. This answers a problem posed by Kohn and Ranestad…

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times…

For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P…

We address the following generalization $P$ of the Lowner-John ellipsoid problem. Given a (non necessarily convex) compact set $K\subset R^n$ and an even integer $d$, find an homogeneous polynomial $g$ of degree $d$ such that $K\subset…

We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close…

Let $X$ be a (real or complex) infinite dimensional linear space. We establish conditions on a homogeneous polynomial $P$ on $X$ so that, if $W$ is any finite dimensional subspace of $X$ on which $P$ vanishes, then $P$ vanishes on an…

Let $d$ be a nonnegative integer, and let $P \subset \mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\binom{2d+1}{d},$…

Let $F\in\mathbb{C}[x,y,z]$ be a constant-degree polynomial,and let $A,B,C\subset\mathbb C$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian product $A\times B\times C$, unless $F$ has a…

We compute the maximum number of rational points at which a homogeneous polynomial can vanish on a weighted projective space over a finite field, provided that the first weight is equal to one. This solves a conjecture by Aubry, Castryck,…