{"id":"hexagon/2611.00040","pdf":"https://hexagonmath.org/pdf/2611.00040v1","url":"https://hexagonmath.org/abs/2611.00040v1","tags":["math.AC","msc:12E05","msc:11T06","msc:13P15","Casas-Alvero conjecture","Hasse derivatives","bad primes","polynomials over finite fields","recycled roots","reduction modulo p"],"score":0,"title":"Recycled roots under reduction and descent in the Casas-Alvero problem","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-03","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}