On hardness of computing analytic Brouwer degree

Abstract

We prove that counting the analytic Brouwer degree of rational coefficient polynomial maps in $\mathrm{Map}({\mathbb{C}}^d,{\mathbb{C}}^d)$ -- presented in degree-coefficient form -- is hard for the complexity class $♯P$, in the following sense: if there is a randomized polynomial time algorithm that counts the Brouwer degree correctly for a good fraction of all input instances (with coefficients of bounded height where the bound is an input to the algorithm), then $P^{♯P}=\mathrm{BPP}$.

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