Recycled roots under reduction and descent in the Casas-Alvero problem
Source: hexagonmath.org/abs/2611.00040v1
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…
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