Extensions of polynomial plank covering theorems

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.

Lemma 4.1 is corrected after the official publication, fortunately, not affecting the other results