A Pólya “shire” theorem for functions with algebraic singularities

[摘要] The classical “shire” theorem of Pólya is proved for functions with algebraic poles, in the sense of L. V. Ahlfors. A functionf(z)is said to have an algebraic pole atz0provided there is a representationf(z)=∑k=−N∞ak(z−z0)k/p+A(z), wherepandNare positive integers andA(z)is analytic atz0. Forp=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.

[学科分类] 数学（综合）