math.NT — Number Theory

Exact Intervals and Arithmetic Structure in Fuchs' Extremal Problem

Contributed by Zhenyu Sun

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, 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 $1$ through $10$ and for $12$, and a two-value alternative at $11$. Subset counting gives a quadratic upper bound in general. The interval formulas and explicit constructions show that $2$, $3$ and $4$ are strict local maxima; the formula on $(2,3)$ has a unique minimum. A Pell construction gives an upper bound on $(4,5)$, 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 $17/4$ for every degree bound from $61$ to $80$.

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