{"id":"hexagon/2611.00126","pdf":"https://hexagonmath.org/pdf/2611.00126v1","url":"https://hexagonmath.org/abs/2611.00126v1","tags":["math.CA","math.CV","msc:11C08","msc:11D72","msc:42A05","msc:11Y50","msc:30C10","Fuchs extremal problem","Prouhet-Tarry-Escott problem","exact lattice enumeration"],"score":0,"title":"Exact Intervals and Arithmetic Structure in Fuchs' Extremal Problem","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Zhenyu Sun"],"reviews":0,"subject":"math.NT","summary":"We determine Fuchs' weighted unit-circle extremum for monic integer polynomials for $0<\\lambda\\le3$. The proof combines a fractional difference energy with a sharp rearrangement inequality and an explicit gap for nonnegative integer sequences that are not interval indicators. We obtain the finite-degree minimum on $(1,2)$ at every degree and on $(2,3)$ at every degree at least five. Every minimizing sequence on these intervals eventually consists, up to monomial factors, of Dirichlet polynomials and products of two adjacent Dirichlet polynomials, respectively. At integer parameters,…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex"],"discussed":0,"published":"2026-11-05","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}