Year: 2026
Type: preprint
Abstract: We develop a new method which allows us to estimate, with a power saving, sums of the M{ö}bius function over function fields ${\mathbb F}_p(X)$. The sums run over monic polynomials with all other coefficients in short intervals, of length below the Pólya--Vinogradov threshold $p^{1/2}$. 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.
Source: arXiv (Cornell University)
Institutions: Université Paris-Saclay, Laboratoire de Mathématiques d'Orsay
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223206587