Combinatorics Seminar - Roman Karasev

16:15-18:00

Speaker: Roman Karasev

Title: Covering by planks and avoiding zeros of polynomials

Abstract: We note that the recent polynomial proofs of (particular
cases 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. After that we establish
polynomial analogs of the Bang theorem by explaining how to find a point
in the unit ball sufficiently far from the zero set of a given
polynomial. As a corollary of these results, we establish a conjecture
of Jiang and Polyanskii about covering a sphere by spherical segments
generalizing the zone conjecture of Fejes Tóth.

Joint work with Alexey Glazyrin and Alexander Polyanskii.