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

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