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The classical shire theorem of Pólya is proved for functions with algebraic poles, in the sense of L. V. Ahlfors. A function f(z) is said to have an algebraic pole at z0 provided there is a representation f(z)=∑k=−N∞ak(z−z0)k/p+A(z), where p and N are positive integers and A(z) is analytic at z0. For p=1, the proof given reduces to an entirely new proof of the shire theorem. New quantitative results are given on how zeros of the successive derivatives migrate to the final set

A Pólya shire theorem for functions with algebraic singularities

A Pólya shire theorem for functions with algebraic singularities