On hardness of computing analytic Brouwer degree

TL;DR

This paper demonstrates that computing the analytic Brouwer degree for polynomial maps with rational coefficients is computationally hard, specifically -sharp P-hard, implying significant complexity-theoretic implications for randomized algorithms.

Contribution

It establishes the -sharp P-hardness of counting the Brouwer degree for polynomial maps with rational coefficients, linking it to fundamental complexity class separations.

Findings

Counting Brouwer degree is -sharp P-hard.

Efficient randomized algorithms for this problem would imply -sharp P = BPP.

The result connects topological degree computation with complexity theory.

Abstract

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 $\operatorname{P}^{\operatorname{\sharp P}} =\operatorname{BPP}$.