ON CERTAIN PLANE CURVES WITH MANY INTEGRAL POINTS

FERNANDO RODRIGUEZ VILLEGAS AND JOSÉ FELIPE VOLOCH

Abstract. We define a sequence of polynomials P_{d} ∈ ℤ[x, y], such that P_{d} is absolutely irreducible, of degree d, has low height, and has at least d^{2} +2d+3 integral solutions to P_{d}(x, y) = 0. We know of no other nontrivial family of polynomials of increasing degree with as many integral solutions in terms of their degree.