math.NT — Number Theory

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

Contributed by David Niedbala Giraudin

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 $p$.