Density of rational points on diagonal quartic surfaces

Abstract

Let $a,b,c,d$ be nonzero rational numbers whose product is a square, and let $V$ be the diagonal quartic surface in ${\mathbb{P}}^3$ defined by $a{x}^4+b{y}^4+c{z}^4+d{w}^4=0$. 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.