  We give an upper bound on character sums of the resultant over pairs of monic
square-free polynomials of given degrees, answering a question of Ellenberg and
Shusterman in the quadratic case. Our approach is topological: we compute the
homology of braid groups on multi-punctured planes and prove a vanishing range
for the homology of mixed braid groups with rank-1 local coefficients
associated to characters of finite fields. Our method involves constructing a
cellular stratification for configuration spaces of multi-punctured planes and
relating their twisted homology with more general exponential coefficients to
the cohomology of certain bimodules over quantum shuffle algebras.
