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

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 $m$.