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A characterization of sums of $ 2n$th powers of global meromorphic functions

Author: Jesús M. Ruiz
Journal: Proc. Amer. Math. Soc. 109 (1990), 915-923
MSC: Primary 32C25; Secondary 12D15
MathSciNet review: 1023347
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Abstract: Let $ M$ be a real analytic manifold. In this note we prove

Theorem. Let $ X$ be a compact analytic set of $ M$ and $ \sum $ its singular locus. Then, a meromorphic function $ h$ on $ X$ is a sum of $ 2n$-th powers of meromorphic functions if and only if, for every analytic curve $ \sigma :\left( { - \varepsilon ,\varepsilon } \right) \to X$ not contained in $ \sum $, it holds $ h \circ \sigma = a{t^m} + \cdots $, with $ a > 0$ and $ 2n$ dividing $ m$.

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