Abstract
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 d, compile a list of bad primes for that degree (namely, those primes p for which the conjecture fails in degree d and characteristic p) and then deduce the conjecture for all degrees of the form \(dp^\ell \), \(\ell \in \mathbb {N}\), where p is a good prime for d. In this paper, we calculate certain distinguished monomials appearing in the resultant \(R(f,H_i(f))\) and obtain a (non-exhaustive) list of bad primes for every degree \(d\in \mathbb {N}\setminus \{0\}\).
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Acknowledgements
1. The fact that the monomial \(\left( 1-\left( {\begin{array}{c}d\\ i\end{array}}\right) \right) ^{d-i} a_{d-i}^d\) appears in \(R_i\) and is the only pure power appearing there was first proved by Rosa de Frutos in her Ph.D. thesis [3], Proposition 2.2.1, page 17. We thank César Massri for pointing this out to us. 2. We thank both referees for corrections and suggestions that helped improve the paper.
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Dedicated to Professor Osamu Saeki on the occasion of his sixtieth birthday.
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Schaub, D., Spivakovsky, M. On the set of bad primes in the study of the Casas–Alvero conjecture. Res Math Sci 11, 31 (2024). https://doi.org/10.1007/s40687-024-00444-z
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DOI: https://doi.org/10.1007/s40687-024-00444-z