Random Diophantine equations of large degree

TL;DR

This work analyzes the probability that a randomly chosen high-degree homogeneous polynomial with coefficients in {−1,1} defines a hypersurface in projective space that possesses a rational point. The authors develop a moment-method framework, studying the first and second moments of rational points in carefully selected finite point sets and deriving sharp combinatorial bounds, local solubility criteria, and parity-control lemmas. They prove that in the Fano-type regime $n \,\ge\, d + \log d$ the density $r_{d,n}$ of hypersurfaces with rational points tends to 1 at rate $O(d^{-1/4})$, and, outside a density-1 exceptional set of degrees, in the wider range $n \ge d^{1/2}\log d$ one again gets $r_{d,n} = 1 + O(d^{-1/2})$. The results rely on binomial-coefficient identities, Vandermonde convolution, and a careful treatment of local solubility via adèles, indicating that most high-degree random hypersurfaces carry rational points even in the general-type regime.

Abstract

Among the set of hypersurfaces of degree $d$ and dimension $\ell$ defined by the vanishing of a homogeneous polynomial with coefficients $\pm 1$, we investigate the probability that a hypersurface contains a rational point as $d$ and $\ell$ tend to infinity.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2