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We prove that some holomorphic functions on the moduli space of tori haveonly simple zeros. Instead of computing the derivative with respect to themoduli parameter $\tau$, we introduce a conceptual proof by applyingPainlev\'{e} VI\ equation. As an application of this simple zero property, weobtain the smoothness of all the degeneracy curves of trivial critical pointsfor some multiple Green function.

Simple zero property of some holomorphic functions on the moduli space of tori