Random Diophantine equations of large degree

TL;DR

This paper investigates the likelihood that high-degree, large-dimension hypersurfaces defined by homogeneous polynomials with coefficients ±1 contain rational points, as both degree and dimension grow large.

Contribution

It introduces a probabilistic analysis of rational points on hypersurfaces with coefficients ±1 in high degree and dimension regimes.

Findings

Probability tends to zero for large degree and dimension

Asymptotic behavior of rational points on these hypersurfaces

Insights into Diophantine equations with restricted coefficients

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.