A Quantitative Hasse Principle for Weighted Quartic Forms

TL;DR

This paper establishes an asymptotic formula for the number of integral solutions to a class of weighted quartic forms using the Hardy-Littlewood method, improving upon previous results under certain conditions.

Contribution

It introduces a new approach to count solutions of weighted quartic forms, extending the applicability of the Hardy-Littlewood method beyond Birch's traditional bounds.

Findings

Asymptotic formula derived for integral solutions

Results hold under conditions s_1 ≥ 2 and 2s_1 + s_2 > 8

Improves upon previous methods for weighted quartic forms

Abstract

We derive, via the Hardy-Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of non-singular local solubility. Our polynomials $F({\mathbf x},{\mathbf y}) \in \mathbb{Z}[x_1,\ldots,x_{s_1},y_1,\ldots,y_{s_2}]$ satisfy the condition that $F(\lambda^2 {\mathbf x}, \lambda {\mathbf y}) = \lambda^4 F({\mathbf x},{\mathbf y})$. Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in many circumstances the expected asymptotic formula holds when $s_1 \ge 2$ and $2s_1 + s_2 > 8$.