Stratification for Multiplicative Character Sums

Abstract

Abstract We prove a stratification result for certain families of n-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of r multiplicative characters evaluated at rational functions, and the families (with nr parameters) are obtained by allowing each of the r rational functions to be replaced by an “offset”, that is, a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least j⌊(r − 1)/2(n − 1)⌋ for 1 ≤ j ≤ n − 1 and ⌈nr/2⌉ for j = n. This result is used to obtain multivariate Burgess bounds in joint work with Lillian Pierce.