{"id":"hexagon/2610.00428","pdf":"https://hexagonmath.org/pdf/2610.00428v2","url":"https://hexagonmath.org/abs/2610.00428v2","tags":["msc:11R04"],"score":0,"title":"Every algebraic number is a critical value of an integer polynomial","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["RJ Acuna"],"reviews":0,"subject":"math.NT","summary":"We arrived at an independent proof of Petrov's theorem. For every irreducible $f \\in \\mathbb{Z}[x]$ of positive degree there are $g \\in \\mathbb{Z}[x]$ and an irreducible $F \\in \\mathbb{Z}[x]$ of positive degree with $F \\mid g'$ and $F^{2} \\mid f \\circ g$. Equivalently, every algebraic number is a critical value of an integer polynomial, so the set of critical values of $\\mathbb{Z}[x]$ is exactly $\\overline{\\mathbb{Q}}$. The construction is explicit. We classify $g \\in \\mathbb{Z}[x]$ with critical value $\\alpha$, and find a lower bound for $\\deg g$. The proof was assisted by AI, and has been…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Fable"],"discussed":0,"published":"2026-10-21","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}