The Hasse principle for homogeneous polynomials with random coefficients over thin sets

In this paper, we investigate the solubility of homogeneous polynomial equations. The work of Browning, Le boudec, Sawin [3] shows that almost all homogeneous equations of degree $d\geq 4$ in $d+1$ or more variables satisfy the Hasse principle, and in particular that a positive portion possess a non-trivial integral solution. Our main result, when combined with our sequel joint work with H.Lee and S.Lee, shows that such a conclusion remains true even when the coefficients of homogeneous polynomials are constrained by a polynomial condition under a modest condition on the number of variables. To be precise, let $d$ and $n$ be natural numbers. Let $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^N$ denote the Veronese embedding with $N=\binom{n+d-1}{d}$, defined by listing all the monomials of degree $d$ in $n$ variables using the lexicographical ordering. Let $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product. For a non-singular form $P\in \mathbb{Z}[\boldsymbol{x}]$ in $N$ variables of degree $k\geq 2,$ consider a set of integer vectors $\boldsymbol{a}\in \mathbb{Z}^N$, defined by $$\mathfrak{A}(A;P)=\{\boldsymbol{a}\in \mathbb{Z}^N|\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\}.$$ We confirm that when $d\geq 4$, $n$ is sufficiently large in terms of $d$, and $k\leq d,$ the proportion of integer vectors $\boldsymbol{a}\in \mathbb{Z}^N$ in $\mathfrak{A}(A;P)$, whose associated equations $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle=0$ satisfy the Hasse principle, converges to $1$ as $A\rightarrow \infty$. We make explicit a lower bound on $n$ guaranteeing this conclusion. In particular, we show that when $d\geq 14$ it suffices to take $n\geq 32d+17$.