Year: 2026
Type: preprint
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 formulas that work over any field with an absolute value and in every characteristic. Over a complete non-archimedean field 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, identities for generalized binomial series, and elementary non-archimedean analysis.
Source: arXiv (Cornell University)
Authors: Bland, Jason, Garibaldi, Skip, Rosenberg, Joel
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162467448