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 $F(\mathbf{x},\mathbf{y})\in \mathbb{Z}[x_1,\dots ,x_{s_1},y_1,\dots ,y_{s_2}]$ 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 $2s_1+s_2>8$.