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

We consider polynomial approximations of $\overline{z}$ $\bar{z}$ to better understand the torsional rigidity of polygons. Our main focus is on low degree approximations and associated extremal problems that are analogous to Pólya’s Conjecture for torsional rigidity of polygons. We also present some numerics in support of Pólya’s Conjecture on the torsional rigidity of pentagons.