A Quantitative Hasse Principle for Weighted Quartic Forms

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 $2s_1 + s_2 > 8$.