  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 $n\geq 32d+17$.

Number Theory