On weighted forms in many variables

Abstract

In this paper, we introduce several novel approaches utilizing the circle method to obtain the asymptotic formula for the number of integral points of bounded height lying on a hypersurface in a weighted projective space. Let $F(\mathbf{x};\mathbf{y})$ be a given weighted form of degree $d$ in variables $\mathbf{x}\in {\mathbb{R}}^{s_1}$ and $\mathbf{y}\in {\mathbb{R}}^{s_2}$, where variables $\mathbf{x}$ and $\mathbf{y}$ have weights $w_1$ and $w_2$ with $w_1<w_2,\left(w_1,w_2\right)=1$, and $d>w_1{w}_2$. Write $$R_F(P):=\#\left\{(\mathbf{x};\mathbf{y})\in {\mathbb{Z}}^{s_1+s_2}:F(\mathbf{x};\mathbf{y})=0,|\mathbf{x}|\le P^{w_1/d},|\mathbf{y}|\le P^{w_2/d}\right\}.$$ In particular, we show that whenever $$s_1+s_2-{\sigma }_F>\left(1+\frac{w_2}{w_1}\right)\frac{d}{w_1}{2}^{d/w_1},$$ where ${\sigma }_F$ is the dimension of the affine singular locus of $F$, the quantity $R_F(P)$ has the expected asymptotic formula, that is $$R_F(P)=c{P}^{s_1{w}_1/d+s_2{w}_2/d-1}+o\left(P^{s_1{w}_1/d+s_2{w}_2/d-1}\right),$$ where $c$ is the product of local densities. Furthermore, the constant $c$ is positive whenever $F(\mathbf{x};\mathbf{y})=0$ has a nonsingular solution over $\mathbb{R}$ and ${\mathbb{Q}}_p$ for every prime $p$. As a corollary, we verify the integral Hasse principle for the quasi-smooth hypersurface defined by $F(\mathbf{x};\mathbf{y})=0$ in a weighted projective space of sufficiently large dimensions.

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