Ultra-short sums of trace functions

Abstract

We generalize results of Duke, Garcia, Hyde, Lutz and others on the distribution of sums of roots of unity related to Gaussian periods to obtain equidistribution of similar sums over zeros of arbitrary integral polynomials. We also interpret these results in terms of trace functions, and generalize them to higher rank trace functions.

v2: 22 pages, 10 figures: correct assumptions in the case of higher-rank trace functions