Plane curves in boxes and equal sums of two powers
  Given an absolutely irreducible ternary form $F$, the purpose of this paper is to produce better upper bounds for the number of integer solutions to the equation F=0, that are restricted to lie in very lopsided boxes. As an application of the main result, a new paucity estimate is obtained for equal sums of two like powers. 
T.D. Browning and D.R. Heath-Brown