Covering by planks and avoiding zeros of polynomials

TL;DR

This paper extends polynomial methods to covering problems on spheres and Euclidean balls, providing new bounds and generalizations of classical theorems like Bang's and the zone conjecture.

Contribution

It introduces polynomial analogs of covering theorems, including a tight bound for polynomials on the Euclidean ball and a strengthened version of the zone conjecture.

Findings

Existence of points at distance at least 1/n from polynomial zeros in the unit ball

Polynomial bounds generalize Bang's theorem for Euclidean balls

Strengthening of the Fejes Tóth zone conjecture with a minimum sum of widths

Abstract

We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball we show that for every real nonzero $d$-variate polynomial $P$ of degree $n$, there exists a point in the unit $d$-dimensional ball at distance at least $1/n$ from the zero set of the polynomial $P$. Using the polynomial approach, we also prove the strengthening of the Fejes T\'oth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least $\pi$.