Metadata
bibkey: kraussimanek2024polynomialcontent
authors: A. Kraus and B. Simanek
year: 2024
title: New Perspectives on Torsional Rigidity and Polynomial Approximations of z-bar
doi: 10.1007/s10659-024-10087-4
url: https://arxiv.org/abs/2309.16450
claim: Conjecture 2.1 asserts that for every polynomial degree there is a polygon side count for which the unit-area polynomial content has no maximizer.
strata_touched:
- D5/S3/Analytic/Torsion/PlanarFanPolynomialApproximation
- D5/S3/Analytic/Torsion/PolynomialContentNoMaximizer
license: citation-only
triage: anchor
Verified locator
DOI: https://doi.org/10.1007/s10659-024-10087-4
Source: https://arxiv.org/abs/2309.16450
Source statement
arXiv:2309.16450v1, p. 3, Conjecture 2.1:
For each N ∈ ℕ, there is an n ∈ ℕ so that among all n-gons with area 1, ρ_N has no maximizer.
Section 1, p. 2, characterizes the polynomial content:
Notice that ρ_N(Ω) is the square of the distance from z̄ to P_N in L²(Ω, dA)
Here P_N is the space of polynomials of degree at most N. Section 2 considers simply connected Jordan regions in the complex plane. Theorem 2.2 states:
Among all hexagons with area 1, ρ₁ and ρ₂ have no maximizer.
The Lean polygon index n represents n + 3 sides. The area is planar volume; the polynomial content is the infimum of the squared integral approximation errors over complex polynomials of degree at most N.
Scope
The star construction with m = max(3, N + 2) gives unit-area 2m-gons with unbounded degree-N polynomial content. The rotational symmetry cancels the complex moments through order N + 1, and stretching increases the polar second moment. The minimal number of sides is a separate question.
Literature boundary
The arXiv v1 source is the statement source. The J. Elasticity article body and the full Baylor dissertation of Kraus are unread; whether either gives a settlement for N ≥ 3 is ASSUMED-UNVERIFIED. Literature completeness is ASSUMED-UNVERIFIED. The source’s degree-one and degree-two hexagon result is not claimed as new.