PreprintCC BY 4.0arXiv
Bounds for the binary quartic Weyl sum
Abstract
We prove that ∑1≤ x,y≤ N e(αF(x,y)) ≪F,α,εN5/3+ε for every ε>0, real quadratic irrational α, and integral binary quartic form F that is not the fourth power of a linear form over ℂ. The proof uses the two-dimensional q-analogue of van der Corput's method, following Heath-Brown's work in the one-variable case. The heart of our argument rests on new bounds for complete binary cubic Weyl sums with linear terms, which retain the dependence on the cubic discriminant and the linear coefficients.
Related
- Betti bounds for spaces of curves on varieties and Manin's conjecture for quartic del Pezzo surfaces
- Weyl Bounds for GL2× GL2 Rankin--Selberg L-functions via Short Spectral Moments
- Improved Weyl bounds on short intervals
- Berry-Esseen Bounds for the Number of Real Zeros of Gaussian Weyl Polynomials
- Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes
BibTeX
@misc{galiote2026bounds,
title = {{Bounds for the binary quartic Weyl sum}},
author = {Daniel Flores Galiote},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03583},
eprint = {2611.03583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03583},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounds for the binary quartic Weyl sum
AU - Daniel Flores Galiote
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03583
UR - https://arxiv.org/abs/2611.03583
AB - We prove that ∑_1≤ x,y≤ N e(αF(x,y)) ≪_F,α,εN^5/3+ε for every ε>0, real quadratic irrational α, and integral binary quartic form F that is not the fourth power of a linear form over ℂ. The proof uses the two-dimensional q-analogue of van der Corput's method, following Heath-Brown's work in the one-variable case. The heart of our argument rests on new bounds for complete binary cubic Weyl sums with linear terms, which retain the dependence on the cubic discriminant and the linear coefficients.
ER -