PreprintarXiv licencearXiv
Short character sums of inhomogeneous polynomials
Abstract
Let p be a prime. We prove nontrivial bounds on short sums of Dirichlet characters mod p evaluated at a class of polynomials, not necessarily homogeneous, in n variables and of degree k. For large n, we further achieve nontrivial bounds for sums over boxes with side-lengths as short as p1/(k−1)+ε, which breaks past the Burgess barrier of p1/4+ε as soon as k≥ 6. In the proof, we develop a new variation of the Burgess amplification method that reduces the problem to bounding additive character sums. This is the first case in which a Burgess-type method succeeds in the inhomogeneous setting.
Related
- Large value estimates for Dirichlet polynomials and trilinear sums over short intervals
- Reciprocity formulas for generalized Dedekind character sums
- A1 - Coulomb branches over finite fields and Selberg Character sums
- Character sums on an oriented singer conic and explicit Ramanujan double covers
- Tridimensional character sums with polynomial arguments and applications
BibTeX
@misc{chu2026short,
title = {{Short character sums of inhomogeneous polynomials}},
author = {Rena Chu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04092},
eprint = {2609.04092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04092},
}
RIS
TY - UNPB
TI - Short character sums of inhomogeneous polynomials
AU - Rena Chu
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04092
UR - https://arxiv.org/abs/2609.04092
AB - Let p be a prime. We prove nontrivial bounds on short sums of Dirichlet characters mod p evaluated at a class of polynomials, not necessarily homogeneous, in n variables and of degree k. For large n, we further achieve nontrivial bounds for sums over boxes with side-lengths as short as p^1/(k−1)+ε, which breaks past the Burgess barrier of p^1/4+ε as soon as k≥ 6. In the proof, we develop a new variation of the Burgess amplification method that reduces the problem to bounding additive character sums. This is the first case in which a Burgess-type method succeeds in the inhomogeneous setting.
ER -