Zeros of analytic functions, with or without multiplicities

Details Zeros of analytic functions, with or without multiplicities Zeros of analytic functions, with or without multiplicities

2011-09-08

arxiv.org/abs/1109.1772v2

Math. Ann. 352 (2012), No. 3, 625-641

Mathematics

Complex Variables

This is a retitled and slightly revised version of my paper arXiv:1004.3591

Scientific paper

10.1007/s00208-011-0655-2

The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound on the number of distinct zeros of the polynomial $abc$ in terms of the degrees of $a$, $b$ and $c$. We extend this to general analytic functions living on a reasonable bounded domain $\Omega\subset\mathbb C$, rather than on the whole of $\mathbb C$. The estimates obtained are sharp, for any $\Omega$, and a generalization of the original result on polynomials can be recovered from them by a limiting argument.

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