Density of rational points on diagonal quartic surfaces

TL;DR

This paper proves that for certain diagonal quartic surfaces with a rational point outside lines and coordinate planes, the rational points are dense in both algebraic and real topologies.

Contribution

It establishes the density of rational points on diagonal quartic surfaces under specific conditions, advancing understanding of rational points on higher-degree surfaces.

Findings

Rational points are dense in the Zariski topology.

Rational points are dense in the real analytic topology.

Density holds if a rational point exists outside lines and coordinate planes.

Abstract

Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in PP^3 defined by ax^4+by^4+cz^4+dw^4=0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic topology.