A note on the Casas-Alvero Conjecture
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929708053037056 |
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| author | Schaub, Daniel Spivakovsky, Mark |
| author_facet | Schaub, Daniel Spivakovsky, Mark |
| contents | The Casas--Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H\_i(f)$ is a power of a linear polynomial. Let $f=x^d+a\_1x^{d-1}+\cdots+a\_1x \in K[a\_1,\ldots,a\_{d-1}][x]$ and let $R\_i = Res(f,H\_i(f))\in K[a\_1,\ldots,a\_{d-1}]$ be the resultant of $f$ and $H\_i(f)$, $i \in \{1,\ldots,d-1\}$. The Casas-Alvero Conjecture is equivalent to saying that $R\_1,\ldots,R\_{d-1}$ are ``independent'' in a certain sense, namely that the height $ht(R\_1,\ldots,R\_{d-1})=d-1$ in $K[a\_1,\ldots,a\_{d-1}]$. In this paper we prove a very partial result in this direction : if $i \in \{d-3,d-2,d-1\}$ then $R\_i \notin \sqrt{(R\_1,\ldots,\breve{R\_i},\ldots,R\_{d-1}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_08742 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on the Casas-Alvero Conjecture Schaub, Daniel Spivakovsky, Mark Commutative Algebra Algebraic Geometry The Casas--Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H\_i(f)$ is a power of a linear polynomial. Let $f=x^d+a\_1x^{d-1}+\cdots+a\_1x \in K[a\_1,\ldots,a\_{d-1}][x]$ and let $R\_i = Res(f,H\_i(f))\in K[a\_1,\ldots,a\_{d-1}]$ be the resultant of $f$ and $H\_i(f)$, $i \in \{1,\ldots,d-1\}$. The Casas-Alvero Conjecture is equivalent to saying that $R\_1,\ldots,R\_{d-1}$ are ``independent'' in a certain sense, namely that the height $ht(R\_1,\ldots,R\_{d-1})=d-1$ in $K[a\_1,\ldots,a\_{d-1}]$. In this paper we prove a very partial result in this direction : if $i \in \{d-3,d-2,d-1\}$ then $R\_i \notin \sqrt{(R\_1,\ldots,\breve{R\_i},\ldots,R\_{d-1}}$. |
| title | A note on the Casas-Alvero Conjecture |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2312.08742 |