Random Diophantine equations of large degree

2025/10/30 by Browning, Tim, Sawin, Will
#11D45 (11G50 #11P21 #14G05) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.26191

Abstract

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

Citations

• The Hasse principle for random Fano hypersurfaces
• On the distribution of reducible polynomials

Related

• Statistics on Hilbert's Sixteenth Problem
• Distribution of rational points on random Fano hypersurfaces
• The density of rational points on non-singular hypersurfaces, I
• Density of Rational Points Near Flat/Rough Hypersurfaces
• Random Diophantine Equations in the Primes II