Plane curves in boxes and equal sums of two powers

Abstract

Let F ∈ Z[x1, x2, x3] be an absolutely irreducible form of degree d, producing a plane curve in P. The central aim of this paper is to analyze the density of rational points on such curves, which are contained in boxes with unequal sides. We shall see below how such considerations may be used to obtain new paucity results for equal sums of two powers. Suppose that P = (P1, P2, P3) for fixed real numbers 1 ≤ P1 ≤ P2 ≤ P3, say. Then we define