Root of the generic cubic as a power series in the discriminant

Abstract

An observation of J-P. Serre implies that the generic monic cubic polynomial, unique among generic monic polynomials of degree at least two, has a root that is a power series in the discriminant; Serre asked for a formula. We give one that works over any field with an absolute value and in every characteristic. Over a complete non-archimedean field of residue characteristic different from 3 we identify the root intrinsically: it is the isolated root, the one farthest from the others. We also answer the next case of Serre's question, computing explicitly the distinguished ramified quadratic factor of the generic monic quartic. The methods combine Hensel's lemma, Lagrange inversion, and elementary non-archimedean analysis.

v3 improves the exposition in the last two sections; v2 adds a result on complete non-archimedean fields, results about real numbers moved to a companion paper, see arXiv:2606.26589