Year: 2026
Type: preprint
Abstract: We prove that $$ \sum_{1\le x,y\le N} e(αF(x,y)) \ll_{F,α,ε}N^{5/3+ε} $$ 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 $\mathbb{C}$. 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.
Source: arXiv (Cornell University)
Author: Galiote, Daniel Flores
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225156362