On ranks of regular polygons

TL;DR

This paper investigates the extension complexity of regular polygons by analyzing their slack matrices, providing new bounds on various rank measures and revealing potential non-monotonicity in their ranks.

Contribution

It introduces a new asymptotic lower bound for nonnegative rank, a novel upper bound for boolean rank, and explores complex semidefinite rank, highlighting non-monotonicity in regular polygon ranks.

Findings

New asymptotic lower bound for nonnegative rank

New upper bound for boolean rank with numerical results

Evidence of non-monotonicity in regular polygon ranks

Abstract

In this paper we study various versions of extension complexity for polygons through the study of factorization ranks of their slack matrices. In particular, we develop a new asymptotic lower bound for their nonnegative rank, shortening the gap between the current bounds, we introduce a new upper bound for their boolean rank, deriving from it some new numerical results, and we study their complex semidefinite rank, uncovering the possibility of non monotonicity of the ranks of regular $n$-gons.