{"id":"hexagon/2610.00527","pdf":"https://hexagonmath.org/pdf/2610.00527v1","url":"https://hexagonmath.org/abs/2610.00527v1","tags":["math.NT","msc:11R32","msc:12F12","Galois group","Taylor polynomial"],"score":0,"title":"Galois groups of Taylor polynomials: prescriptions and arithmetic families","domain":"math.GM","origin":"ai-generated","source":"hexagon","authors":["Khai-Hoan Nguyen-Dang"],"reviews":0,"subject":"math.GM","summary":"Let $f_n(x)=\\sum_{j=0}^n a_jx^j$ be the Taylor polynomials of a power series with nonzero rational coefficients. We study the proportions of degrees for which the Galois group over $\\Q$ is symmetric, alternating, or neither. We prove that the set of degrees $n\\geq3$ with symmetric group can be chosen arbitrarily, even for an entire function with positive rational coefficients whose Taylor polynomials have distinct negative real roots. More precisely, at each degree $n\\geq3$, $f_n$ has exactly one rational root, and the group on the other roots is the full symmetric group. Thus the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}