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.
| Subjects: | Rings and Algebras (math.RA) |
| MSC classes: | 12J05, 12D10 |
| Cite as: | arXiv:2605.25992 [math.RA] |
| (or arXiv:2605.25992v1 [math.RA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25992 arXiv-issued DOI via DataCite (pending registration) |
From: Skip Garibaldi [view email]
[v1]
Mon, 25 May 2026 16:11:43 UTC (41 KB)
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