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

TL;DR

This paper proves that for high-dimensional homogeneous polynomials with random coefficients, almost all such polynomials satisfy the Hasse principle as the size of coefficients grows, extending previous results.

Contribution

It establishes that under certain conditions, the proportion of polynomials satisfying the Hasse principle approaches one, using new lattice and geometry of numbers techniques.

Findings

Proportion of polynomials satisfying Hasse principle tends to 1 as coefficients grow.

Valid for forms with degree d ≥ 17 and dimension n > 24d.

Improves previous results by the second author.

Abstract

Let $d$ and $n$ be natural numbers. Let $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^{N}$ denote the Veronese embedding with $N=N_{n,d}:=\binom{n+d-1}{d}$, 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 $P\in \mathbb{Z}[\boldsymbol{x}]$ of degree $k\ (\leq d)$ in $N$ variables, consider a set of integer vectors $\boldsymbol{a}\in \mathbb{Z}^N$, defined by $$\mathfrak{A}(A;P)=\{\boldsymbol{a}\in \mathbb{Z}^N:\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\}.$$ By handling a new lattice problem via the geometry of numbers, we confirm that whenever $n> 24d$ and $d\geq 17,$ the proportion of integer coefficients $\boldsymbol{a}\in \mathfrak{A}(A;P)$, whose associated equation $f_{\boldsymbol{a}}(\boldsymbol{x})=0$ satisfies the Hasse principle, converges to $1$ as $A\rightarrow\infty$. This improves on the recent work of the second author.