The diophantine problem Y ² − X³ = A in a polynomial ring

Dennis Lee Johnson

Vol. 43 (1972), No. 1, 151–155

DOI: 10.2140/pjm.1972.43.151

Abstract

Let C[z] be the ring of polynomials in z with complex coefficients; we consider the equation Y ² − X³ = A, with A ∈ C[z] given, and seek solutions of this with X,Y ∈ C[z] i.e. we treat the equation as a “polynomial diophantine” problem. We show that when A is of degree 5 or 6 and has no multiple roots, then there are exactly 240 solutions (X,Y ) to the problem with deg X ≦ 2 and deg Y ≦ 3.

Mathematical Subject Classification 2000

Primary: 10B15

Secondary: 14H30, 12A20

Milestones

Received: 15 July 1971

Published: 1 October 1972