New Perspectives on Torsional Rigidity and Polynomial Approximations of z-bar

TL;DR

This paper explores polynomial approximations of the complex conjugate function to analyze torsional rigidity in polygons, focusing on low degree approximations and extremal problems related to Polya's conjecture, supported by numerical evidence.

Contribution

It introduces new polynomial approximation techniques for torsional rigidity and investigates extremal problems analogous to Polya's conjecture for polygons.

Findings

Numerical evidence supporting Polya's conjecture for pentagons.

Development of extremal problems related to polynomial approximations.

Insights into torsional rigidity through low degree polynomial approximations.

Abstract

We consider polynomial approximations of z-bar to better understand the torsional rigidity of polygons. Our main focus is on low degree approximations and associated extremal problems that are analogous to Polya's conjecture for torsional rigidity of polygons. We also present some numerics in support of Polya's Conjecture on the torsional rigidity of pentagons.