The Hasse principle for random homogeneous polynomials in thin sets

Abstract

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 $N=N_{n,d}:=\left(\genfrac{}{}{0ex}{}{n+d-1}{d}\right)$, 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 $P\in \mathbb{Z}[\mathbf{x}]$ of degree $k(\le d)$ in $N$ variables, consider a set of integer vectors $\mathbf{a}\in {\mathbb{Z}}^N$, defined by $$\mathfrak{A}(A;P)=\{\mathbf{a}\in {\mathbb{Z}}^N:P(\mathbf{a})=0,\Vert \mathbf{a}{\Vert }_{\infty }\le A\}.$$ By handling a new lattice problem via the geometry of numbers, we confirm that whenever $n>24d$ and $d\ge 17,$ the proportion of integer coefficients $\mathbf{a}\in \mathfrak{A}(A;P)$, whose associated equation $f_{\mathbf{a}}(\mathbf{x})=0$ satisfies the Hasse principle, converges to $1$ as $A\to \infty$. This improves on the recent work of the second author.

23 pages, Mathematika (to appear, Wooley Special Issue), all comments are welcome!