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

David Niedbala Giraudin

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

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

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

For a Casas-Alvero polynomial

f

, the type of

f

is one less than the minimal number of roots of

f

needed to meet every Hasse derivative

H_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

np^e

to degree

n

in characteristic

p

. Consequently, if no Casas-Alvero polynomial of degree

n

and type at most

k

exists over

\overline{\mathbb F}_p

, then none of degree

np^e

and type at most

k

exists in characteristic

0

, for every

e\ge 0

. For

k=2

the hypothesis is equivalent to

p\nmid N_n

for an explicit integer

N_n

, and we show by a finite computation that

N_n\ne0

for

3\le n\le 22

and

n=24

. In particular there is no counterexample of type at most

2

in degree

24

. This gives infinite families of degrees for which the conjecture is not known and every counterexample has type at least

3

: the degrees

5\cdot 8009^e

, the degrees

6p^e

for

21

primes

p

that are bad for degree

6

, the degrees

7p^e

,

8p^e

,

9p^e

and

10p^e

for

13

,

17

,

16

and

103

explicit primes

p

that are bad for these degrees, and the degrees

20p^e

and

24p^e

for all primes

p

outside a finite set (for

20p^e

, including

p=2^{31}-1

). For

k=3

we obtain the first degrees in which every counterexample has type at least

4

: the degrees

6p^e

for

p\in\{983,1901,51313000813080529\}

, the degrees

8\cdot397^e

, the degrees

9p^e

for

p\in\{613,757\}

and the degrees

10p^e

for

13

primes

p

.

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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C. Massri, The Casas-Alvero conjecture for three recycled roots in degree 20 , arXiv:1806.09561.arXiv
D. Schaub, M. Spivakovsky, A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero conjecture , arXiv:2411.13967.arXiv

Version history

v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 02, 2026