Hardy-Littlewood Varieties.

Let V be a variety (for example a hypersurface) defined by integral polynomial equations in $(x_1,\dots,x_n)$. Let $V(\mathbb{Z} )$ be the integral points in V and let

\begin{displaymath}N_V(T) = \big \vert \lbrace x\in \mathbb{Z} ^n \vert x \in V(\mathbb{Z} ), \vert\vert x\vert\vert \le T \rbrace \big \vert \end{displaymath}
 
We say V is a ``Hardy-Littlewood Variety'' if the asymptotic behavior of NV(T) as V NVT $T\to \infty$ is given by a product of local densities which count the number of solutions in V over the real numbers and the p-adic numbers (the exact form of this product of local densities comes from the Hardy-Littlewood method). The known Hardy-Littlewood varieties are ones shown to be so by their method, which requires that n be very large compared to the degrees of the defining equations. However, it appears (from a number of points of view) that to be a Hardy-Littlewood variety is not so restrictive. A test example would be the case of nonsingular cubic forms F in six variables: V p n F
\begin{displaymath}V : \lbrace x\ \vert\ F(x_1,\dots,x_6) = 0 \rbrace \end{displaymath}

(The case of nine is probably Hardy-Littlewood, while four variables is not!) One could test this by determining the expected Hardy-Littlewood asymptotic and comparing it with the experimental counting of NV(T).

 

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