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Exact Intervals and Arithmetic Structure in Fuchs' Extremal Problem

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CA — Classical Analysis and ODEsmath.CV — Complex Variables

Contributed by Zhenyu Sun ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

We determine Fuchs' weighted unit-circle extremum for monic integer polynomials for 0<λ≤30<\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)(1,2) at every degree and on (2,3)(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, Parseval's identity reduces the problem to integer coefficient norms. The lower bound is attained exactly when a multiplicity-free ideal Prouhet--Tarry--Escott pair exists. Classical PTE constructions give exact values for the integers 11 through 1010 and for 1212, and a two-value alternative at 1111. Subset counting gives a quadratic upper bound in general. The interval formulas and explicit constructions show that 22, 33 and 44 are strict local maxima; the formula on (2,3)(2,3) has a unique minimum. A Pell construction gives an upper bound on (4,5)(4,5), and coefficient-norm gaps exclude Dirichlet products uniformly near certain integers. For each fixed degree bound, the extremum, divided by a common gamma factor, is locally the lower envelope of finitely many rational functions. Exact lattice enumeration identifies a nonproduct minimizer on an open interval containing 17/417/4 for every degree bound from 6161 to 8080.

Author comments: 24 pages. Includes exact-arithmetic programs and data.

Provenance statement

The manuscript was revised with OpenAI Codex for academic exposition and LaTeX presentation. Integer and rational computations were checked and reproduced with Python; the accompanying source archive includes the scripts and exact data.

Tools used

OpenAI
Codex

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Version history

  1. v1 · 24 pagesSubmitted by Zhenyu Sun24 pages. Includes exact-arithmetic programs and data.CurrentNov 05, 2026