The local solubility for homogeneous polynomials with random coefficients over thin sets

TL;DR

This paper studies the likelihood that certain algebraic varieties defined by random homogeneous polynomials over bounded integer coefficients are locally soluble everywhere, showing this probability converges to a positive constant under specific conditions.

Contribution

It establishes the asymptotic proportion of locally soluble varieties within a family defined by a polynomial constraint, extending understanding of local solubility for random algebraic varieties.

Findings

Proportion of locally soluble varieties converges to a constant as coefficient bound increases.

If a specific variety admits a smooth rational point, the limiting constant is positive.

Results apply to varieties defined by polynomials with bounded integer coefficients and a nontrivial solution.

Abstract

Let $d$ and $n$ be natural numbers greater or equal to $2$. 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, and $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^N$ denotes the Veronese embedding with $N=\binom{n+d-1}{d}$. Consider a variety $V_{\boldsymbol{a}}$ in $\mathbb{P}^{n-1}$, defined by $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle=0.$ In this paper, we examine a set of these varieties defined by $$\mathbb{V}^{P}_{d,n}(A)=\{ V_{\boldsymbol{a}}\subset \mathbb{P}^{n-1}|\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\},$$ where $P\in \mathbb{Z}[\boldsymbol{x}]$ is a non-singular form in $N$ variables of degree $k$ with $2 \le k\leq C({n,d})$ for some constant $C({n,d})$ depending at most on $n$ and $d$. Suppose that $P(\boldsymbol{a})=0$ has a nontrivial integer solution. We confirm that the proportion of varieties $V_{\boldsymbol{a}}$ in $\mathbb{V}^{P}_{d,n}(A)$, which are everywhere locally soluble, converges to a constant $c_P$ as $A\rightarrow \infty.$ In particular, if there exists $\boldsymbol{b}\in \mathbb{Z}^N$ such that $P(\boldsymbol{b})=0$ and the variety $V_{\boldsymbol{b}}$ in $\mathbb{P}^{n-1}$ admits a smooth $\mathbb{Q}$-rational point, the constant $c_P$ is positive.