Short sums of the Möbius function over function fields

We develop a new method which allows us to estimate, with a power saving, sums of the M{ö}bius function over function fields ${\text{F}}_p(X)$. The sums run over monic polynomials with all other coefficients in short intervals, of length below the Pólya–Vinogradov threshold $p^{1\mathrm{/}2}$. 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 {à} la carte} are introduced to go beyond the Pólya–Vinogradov range, combined with a stratification result of J. Xu (2020) for complete character sums.