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Recycled roots under reduction and descent in the Casas-Alvero problem

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.AC — Commutative Algebra

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

For a Casas-Alvero polynomial ff, the type of ff is one less than the minimal number of roots of ff needed to meet every Hasse derivative H1f,…,Hd−1fH_1f,\dots,H_{d-1}f. We show that the type does not increase under reduction modulo a prime and under the descent from degree npenp^e to degree nn in characteristic pp. Consequently, if no Casas-Alvero polynomial of degree nn and type at most kk exists over F‾p\overline{\mathbb F}_p, then none of degree npenp^e and type at most kk exists in characteristic 00, for every e≥0e\ge 0. For k=2k=2 the hypothesis is equivalent to p∤Nnp\nmid N_n for an explicit integer NnN_n, and we show by a finite computation that Nn≠0N_n\ne0 for 3≤n≤223\le n\le 22 and n=24n=24. In particular there is no counterexample of type at most 22 in degree 2424. This gives infinite families of degrees for which the conjecture is not known and every counterexample has type at least 33: the degrees 5⋅8009e5\cdot 8009^e, the degrees 6pe6p^e for 2121 primes pp that are bad for degree 66, the degrees 7pe7p^e, 8pe8p^e, 9pe9p^e and 10pe10p^e for 1313, 1717, 1616 and 103103 explicit primes pp that are bad for these degrees, and the degrees 20pe20p^e and 24pe24p^e for all primes pp outside a finite set (for 20pe20p^e, including p=231−1p=2^{31}-1). For k=3k=3 we obtain the first degrees in which every counterexample has type at least 44: the degrees 6pe6p^e for p∈{983,1901,51313000813080529}p\in\{983,1901,51313000813080529\}, the degrees 8⋅397e8\cdot397^e, the degrees 9pe9p^e for p∈{613,757}p\in\{613,757\} and the degrees 10pe10p^e for 1313 primes pp.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

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Version history

  1. v1 · 7 pagesSubmitted by David Niedbala GiraudinInitial depositCurrentNov 02, 2026