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

Abstract

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 $d\ge 4$ 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 $\hat{I}{½}_{d,n}:{\mathbb{R}}^n\to {\mathbb{R}}^N$ denote the Veronese embedding with $N=\left(\genfrac{}{}{0ex}{}{n+d-1}{d}\right)$, defined by listing all the monomials of degree $d$ in $n$ variables using the lexicographical ordering. 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. For a non-singular form $P\in \mathbb{Z}[\mathbf{x}]$ in $N$ variables of degree $k\ge 2,$ consider a set of integer vectors $\mathbf{a}\in {\mathbb{Z}}^N$, defined by $$\mathfrak{A}(A;P)=\{\mathbf{a}\in {\mathbb{Z}}^N|P(\mathbf{a})=0,\Vert \mathbf{a}{\Vert }_{\infty }\le A\}.$$ We confirm that when $d\ge 4$, $n$ is sufficiently large in terms of $d$, and $k\le d,$ the proportion of integer vectors $\mathbf{a}\in {\mathbb{Z}}^N$ in $\mathfrak{A}(A;P)$, whose associated equations $⟨\mathbf{a},\hat{I}{½}_{d,n}(\mathbf{x})⟩=0$ satisfy the Hasse principle, converges to $1$ as $A\to \infty$. We make explicit a lower bound on $n$ guaranteeing this conclusion. In particular, we show that when $d\ge 14$ it suffices to take $n\ge 32d+17$.

81 pages, to appear in Proceedings of the London Mathematical Society