Short sums of the M\"obius function over function fields

Abstract

We develop a new method which allows us to estimate, with a power saving, sums of the M{\"o}bius function over function fields $\F_p(X)$. The sums run over monic polynomials with all other coefficients in short intervals, of length below the P\'olya--Vinogradov threshold $p^{1/2}$. A classical result of L.~Stickelberger (1898) reduces this question to bounding character sums with polynomial discriminants which are inhomogeneous multivariate polynomials. Variants of Burgess's method with non-linear shifts {\it {\`a} la carte} are introduced to go beyond the P\'olya--Vinogradov range, combined with a stratification result of J. Xu (2020) for complete character sums.