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

Fox-Neuwirth cells, quantum shuffle algebras, and character sums of the resultant