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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.25992 |
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Table of Contents:
- 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.