Stratification for multiplicative character sums

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”, i.e. 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 $j\lfloor (r-1)\mathrm{/}2(n-1)\rfloor$ for $1\le j\le n-1$and$\lceil nr/2\rceil$for$j=n$.