The local solubility for homogeneous polynomials with random coefficients over thin sets

Abstract

Let $d$ and $n$ be natural numbers greater or equal to $2$. Let $⟨\mathbf{a},\hat{I}{½}_{d,n}(\mathbf{x})⟩\in \mathbb{Z}[\mathbf{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\mathbf{a}$, where $⟨\cdot ,\cdot ⟩$ denotes the inner product, and $\hat{I}{½}_{d,n}:{\mathbb{R}}^n\to {\mathbb{R}}^N$ denotes the Veronese embedding with $N=\left(\genfrac{}{}{0ex}{}{n+d-1}{d}\right)$. Consider a variety $V_{\mathbf{a}}$ in ${\mathbb{P}}^{n-1}$, defined by $⟨\mathbf{a},\hat{I}{½}_{d,n}(\mathbf{x})⟩=0.$ In this paper, we examine a set of these varieties defined by $${\mathbb{V}}_{d,n}^P(A)=\{V_{\mathbf{a}}\subset {\mathbb{P}}^{n-1}|P(\mathbf{a})=0,\Vert \mathbf{a}{\Vert }_{\infty }\le A\},$$ where $P\in \mathbb{Z}[\mathbf{x}]$ is a non-singular form in $N$ variables of degree $k$ with $2\le k\le C(n,d)$ for some constant $C(n,d)$ depending at most on $n$ and $d$. Suppose that $P(\mathbf{a})=0$ has a nontrivial integer solution. We confirm that the proportion of varieties $V_{\mathbf{a}}$ in ${\mathbb{V}}_{d,n}^P(A)$, which are everywhere locally soluble, converges to a constant $c_P$ as $A\to \infty .$ In particular, if there exists $\mathbf{b}\in {\mathbb{Z}}^N$ such that $P(\mathbf{b})=0$ and the variety $V_{\mathbf{b}}$ in ${\mathbb{P}}^{n-1}$ admits a smooth $\mathbb{Q}$-rational point, the constant $c_P$ is positive.

Mathematika (2024)