Research Problems in Function Theory — Problem 4.25

Definitions and mathematical statement

Fix a real parameter $\lambda>0$. Let $\mathcal M$ be the collection of all monic polynomials with integer coefficients, in any finite degree. The constant polynomial $1$ is included, while the zero polynomial is excluded. For $P\in\mathcal M$, define \[ J_\lambda(P)=\int_{-\pi}^{\pi} |1-e^{i\theta}|^{2\lambda}|P(e^{i\theta})|^2\,d\theta, \qquad I(\lambda)=\inf_{P\in\mathcal M}J_\lambda(P). \] The integral is unnormalized Lebesgue integration in the angle variable; thus there is no factor $1/(2\pi)$. The weight is nonnegative, vanishes only at $\theta=0$ modulo $2\pi$, and is integrable for the stated range of $\lambda$.

Determine $I(\lambda)$ as a function of $\lambda$. A sharp answer must give a lower bound valid simultaneously for all degrees and all integer coefficient choices, together with polynomials whose integrals approach that bound. Determine also whether the infimum is attained, and describe extremal polynomials when they exist. The degree is allowed to grow along an approximating sequence; minimizing separately in each fixed degree is only an intermediate step. No restriction is imposed on the locations of the roots, on the signs of the integer coefficients, or on their magnitudes. The monic condition is essential because arbitrary multiplication by small constants would otherwise destroy the intended arithmetic normalization.

Short English statement

Find the least weighted boundary square integral among all monic integer polynomials, with the degree unrestricted.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2304025 (AMR-022-4025), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 4.25. Original source indexed by AMR-022-4025. https://arxiv.org/abs/1809.07200.