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

Let d$d$ and n$n$ be natural numbers greater or equal to 2. Let ⟨a,νd,n(x)⟩∈Z[x]$\langle \bm{a}, \nu _{d,n}(\bm{x})\rangle \in \mathbb {Z}[\bm{x}]$ be a homogeneous polynomial in n$n$ variables of degree d$d$ with integer coefficients a$\bm{a}$, where ⟨·,·⟩$\langle \cdot,\cdot \rangle$ denotes the inner product, and νd,n:Rn→RN$\nu _{d,n}: \mathbb {R}^n\rightarrow \mathbb {R}^N$ denotes the Veronese embedding with N=n+d−1d$N=\binom{n+d-1}{d}$. Consider a variety Va$V_{\bm{a}}$ in Pn−1$\mathbb {P}^{n-1}$, defined by ⟨a,νd,n(x)⟩=0$\langle \bm{a}, \nu _{d,n}(\bm{x})\rangle =0$. In this paper, we examine a set of integer vectors a∈ZN$\bm{a}\in {\mathbb {Z}}^N$, defined by A(A;P)={a∈ZN:P(a)=0,∥a∥∞⩽A},$$\begin{equation*} \mathfrak {A}(A;P)=\lbrace \bm{a}\in {\mathbb {Z}}^N:\ P(\bm{a})=0,\ \Vert \bm{a}\Vert _{\infty }\leqslant A\rbrace, \end{equation*}$$where P∈Z[x]$P\in \mathbb {Z}[{\bm{x}}]$ is a nonsingular form in N$N$ variables of degree k$k$ with 2⩽k⩽C(n,d)$2 \leqslant k\leqslant C({n,d})$ for some constant C(n,d)$C({n,d})$ depending at most on n$n$ and d$d$. Suppose P(a)=0$P(\bm{a})=0$ has a nontrivial integer solution. We confirm that the proportion of integer vectors a∈ZN$\bm{a}\in {\mathbb {Z}}^N$ in A(A)$\mathfrak {A}(A)$, whose associated equation ⟨a,νd,n(x)⟩=0$\langle \bm{a}, \nu _{d,n}(\bm{x})\rangle =0$ is everywhere locally soluble, converges to a constant cP$c_P$ as A→∞$A\rightarrow \infty$. Moreover, for each place v$v$ of Q${\mathbb {Q}}$, if there exists a nonzero bv∈QvN$\bm{b}_v\in {\mathbb {Q}}_v^N$ such that P(bv)=0$P(\bm{b}_v)=0$ and the variety Vbv$V_{\bm{b}_v}$ in Pn−1$\mathbb {P}^{n-1}$ admits a smooth Qv$\mathbb {Q}_v$‐point, the constant cP$c_P$ is positive.