Fuchs's weighted-$L^2$ extremal problem over monic integer polynomials (Problem 4.25)
Statement
For a fixed real $\lambda>0$, determine $\displaystyle\inf_P\int_{-\pi}^{\pi}\bigl|1-e^{i\theta}\bigr|^{2\lambda}\,\bigl|P(e^{i\theta})\bigr|^2\,d\theta$, where $P$ ranges over all monic polynomials with integer coefficients (i.e. integer coefficients and leading coefficient exactly $1$), of every degree. The infimum is a single constant depending on $\lambda$; a natural decomposition is to compute the minimum over monic integer $P$ of each fixed degree $d$ and then let $d\to\infty$. It is part of the question to decide, for a given $\lambda>0$, whether this infimum is positive.
Acceptance. ADVANCES: certified exact degree-$d$ minima (with the extremal monic integer polynomial and an exact lattice-minimization certificate) over a range of degrees at one or more rational $\lambda$ - new data toward the limit. ADVANCES: a proof that the infimum is positive (or is zero) for a specified $\lambda>0$; rigorous two-sided bounds on the constant. FULLY RESOLVES: the exact value of $\inf_P(\cdots)$ as a function of $\lambda$ (or for a specific $\lambda>0$), with the limiting extremal family identified and convergence proved. Numerical estimates without exact/certified bounds do NOT qualify.
Background
Posed by W. H. J. Fuchs; Problem 4.25 in W. K. Hayman & E. F. Lingham, Research Problems in Function Theory (Fiftieth Anniversary Edition, Springer 2019), source of record arXiv:1809.07200, which notes 'the solution would have number-theoretic applications.' The book's Update 4.25 states 'No progress on this problem has been reported to us.' This weighted-$L^2$ (Hardy/Bergman-type) functional is genuinely DISTINCT from the integer-Chebyshev / integer transfinite-diameter circle of problems (Borwein-Erdelyi, Habsieger, Flammang, Pritsker; e.g. the 'monic integer Chebyshev problem', arXiv:1307.5362), which is an $L^\infty$ sup-norm theory and a different extremal quantity. The weight $|1-e^{i\theta}|^{2\lambda}$ is a Fisher-Hartwig / Jacobi-type circle weight; for fixed degree $d$ the integral is an explicit positive-definite integer quadratic (Toeplitz Gram) form in the free coefficients, so the degree-$d$ minimum over monic integer $P$ is a closest-lattice-vector / integer quadratic minimization (LLL + enumeration). Vetted open as of 2026-07-06 ('No progress' marker; solved-signal search across integer-Chebyshev literature confirms this weighted-$L^2$ functional is untreated).
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Hayman & Lingham, Research Problems in Function Theory (New Edition) - Problem 4.25 (Fuchs) | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.