  We prove that counting the analytic Brouwer degree of rational coefficient
polynomial maps in $\operatorname{Map}(\mathbb C^d, \mathbb C^d)$ -- presented
in degree-coefficient form -- is hard for the complexity class
$\operatorname{\sharp P}$, 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
$\operatorname{P}^{\operatorname{\sharp P}} =\operatorname{BPP}$.
