Root of a cubic polynomial as a power series in the discriminant
Rings and Algebras
2026-05-26 v1
Abstract
An observation by J-P. Serre implies that cubic polynomials are unique among generic monic polynomials of degree 2 or higher in that they have a root that is a power series in the discriminant of the polynomial. We provide formulas for this root of a cubic that work in any characteristic. In the special case of a cubic real polynomial with positive discriminant, the series converges and therefore provides an explicit formula for a root; when that polynomial is depressed, the root we provide is the longest root. The proofs are a combination of elementary techniques from algebra, combinatorics, and analysis and employ the notion of a field with an absolute value.
Keywords
Cite
@article{arxiv.2605.25992,
title = {Root of a cubic polynomial as a power series in the discriminant},
author = {Jason Bland and Skip Garibaldi and Joel Rosenberg},
journal= {arXiv preprint arXiv:2605.25992},
year = {2026}
}