Extensions of polynomial plank covering theorems

TL;DR

This paper extends polynomial plank covering theorems to complex polynomials and non-symmetric planks, broadening the scope of existing geometric covering results with new theoretical proofs.

Contribution

It introduces a complex polynomial plank covering theorem for non-homogeneous polynomials and generalizes Ball's theorem to non-symmetric, non-round planks.

Findings

Extended polynomial plank covering theorem to complex, non-homogeneous polynomials

Generalized Ball's complex plank theorem to non-symmetric, non-round planks

Proved a weaker version of the spherical polynomial plank covering conjecture

Abstract

We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths.