A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero Conjecture
| Autor: | Schaub, Daniel, Spivakovsky, Mark |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | The Casas--Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree $n$, compile a list of bad primes for that degree (namely, those primes $p$ for which the conjecture fails in degree $n$ and characteristic $p$) and then deduce the conjecture for all degrees of the form $np^\ell$, $\ell\in \mathbb{N}$, where $p$ is a good prime for $n$. In this paper we give an explicit description of the set of bad primes in any given degree $n$. In particular, we show that if the conjecture holds in degree $n$ then the bad primes for $n$ are bounded above by $\binom{\frac{n^2-n}2}{n-2}!\prod\limits_{i=1}^{n-1}\binom{i+n-2}{n-2}^{\binom{d-i+n-2}{n-2}}$. Comment: arXiv admin note: text overlap with arXiv:2307.05997 |
| Databáze: | arXiv |
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