Proof of the Casas-Alvero conjecture
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915877807456256 |
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| author | Ghosh, Soham |
| author_facet | Ghosh, Soham |
| contents | The Casas-Alvero conjecture states that if $f(X)$ is a monic univariate polynomial of degree $d$ over a characteristic $0$ field $\mathbb{K}$ such that $\gcd(f, f_{i})$ is non-trivial for each $i=1, \dots, d-1$, then $f(X)=(X-α)^d$ for some $α\in \mathbb{K}$. In this paper, we prove the Casas-Alvero conjecture for polynomials of any degree $d\geq 3$ over any characteristic $0$ field, by using Koszul homology. Along the way we show existence of various "almost counterexamples" over $\mathbb{C}$, satisfying mildly weaker hypotheses, using Brouwer degree techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09272 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of the Casas-Alvero conjecture Ghosh, Soham Commutative Algebra Algebraic Geometry 12E05, 13D02, 13D03, 14M10 The Casas-Alvero conjecture states that if $f(X)$ is a monic univariate polynomial of degree $d$ over a characteristic $0$ field $\mathbb{K}$ such that $\gcd(f, f_{i})$ is non-trivial for each $i=1, \dots, d-1$, then $f(X)=(X-α)^d$ for some $α\in \mathbb{K}$. In this paper, we prove the Casas-Alvero conjecture for polynomials of any degree $d\geq 3$ over any characteristic $0$ field, by using Koszul homology. Along the way we show existence of various "almost counterexamples" over $\mathbb{C}$, satisfying mildly weaker hypotheses, using Brouwer degree techniques. |
| title | Proof of the Casas-Alvero conjecture |
| topic | Commutative Algebra Algebraic Geometry 12E05, 13D02, 13D03, 14M10 |
| url | https://arxiv.org/abs/2501.09272 |