# Ultra-short sums of trace functions

## TL;DR

This paper extends the understanding of the distribution of sums of roots of unity and trace functions, providing new equidistribution results for zeros of integral polynomials and higher rank trace functions.

## Contribution

It generalizes previous results on roots of unity distribution to arbitrary integral polynomials and higher rank trace functions, broadening the scope of equidistribution analysis.

## Key findings

- Established equidistribution of sums over zeros of integral polynomials
- Generalized results to higher rank trace functions
- Connected distribution results to trace function theory

## 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.