Monodromies of singularities of the Hadamard and e\~ne product

Abstract

We prove that singularities with holomorphic monodromies are preserved by the Hadamard product. We find an explicit formula for the monodromy of the singularities, and similar formulas for the exponential e\~ne product. Using these formulas we get new direct proofs of classical results and the invariance of some rings of functions by Hadamard and e\~ne product (which is the product of the big Witt ring). This explains the good behavior of the e\~ne product on divisors.