On hardness of computing analytic Brouwer degree

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 $\mathrm{♯P}$, 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 ${\mathrm{P}}^{\mathrm{♯P}}=\mathrm{BPP}$.