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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2605.25992 |
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| _version_ | 1866913161756540928 |
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| author | Bland, Jason Garibaldi, Skip Rosenberg, Joel |
| author_facet | Bland, Jason Garibaldi, Skip Rosenberg, Joel |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25992 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Root of a cubic polynomial as a power series in the discriminant Bland, Jason Garibaldi, Skip Rosenberg, Joel Rings and Algebras 12J05, 12D10 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. |
| title | Root of a cubic polynomial as a power series in the discriminant |
| topic | Rings and Algebras 12J05, 12D10 |
| url | https://arxiv.org/abs/2605.25992 |