Cox-Zucker machine

The Cox-Zucker machine is an algorithm in arithmetic geometry, introduced by David A. Cox and Steven Zucker for studying elliptic surfaces. Given a set of sections of an elliptic surface, it decides whether that set forms a basis, up to torsion, of the surface's Mordell-Weil group. It works for the case where the base curve is isomorphic to the projective line. A section here is a morphism from the base curve to the surface whose composition with the fibration is the identity, which informally picks one point on each fiber. The underlying method appeared in the 1979 article by Cox and Zucker titled Intersection numbers of sections of elliptic surfaces, and the name Cox-Zucker machine was later used by Charles Schwartz in 1984. The name is a near homophone of an obscenity, a coincidence that Cox recalled in a 2021 tribute to Zucker in the Notices of the American Mathematical Society: when the two met as graduate students at Princeton University they noticed the effect of their surnames in alphabetical order and joked that they should write a joint paper.