Multiplicity-Weighted Moments Across a Harmonic Caustic: Closed Forms with Order-Independent Breakpoints
Abstract
While the Jacobian of a planar harmonic mapping keeps a constant sign, the oriented and the multiplicity-weighted surface elements agree up to that sign, and the two natural moment hierarchies coincide. The nearest classification results in the literature are stated under exactly this hypothesis; this paper works past it. Once the Jacobian changes sign, the critical circle acquires a caustic as its image, part of the plane is covered several times with opposite orientations, and the hierarchies part company: integrating a radial weight against the oriented surface element nz lets overlapping sheets cancel, giving the degree-weighted moments Jn, while integrating against |nz| lets every sheet count positively, giving the multiplicity-weighted moments Jn±. These are the classical degree and area formulas for the same map, the second weighted by the multiplicity function (Banach indicatrix). It is the second family, the one the caustic threatens, for which a closed form is no longer automatic. Our main result is that it has one anyway: Jn± is the total variation of the very same explicit primitive whose endpoint difference gives Jn. Crossing the caustic therefore costs only finitely many extra evaluation points, and they are the same points for every order n; for a monotone profile there is at most one interior breakpoint, at the critical radius itself. The family carrying this is the two-monomial planar harmonic mapping F(w)=w+iCw̅k, w=p(u)eiv, arising as the planar projection of a Fourier-perturbed surface. On the degree-weighted side Jn has a boundary-only closed form for every n≥ 0, the zeroth being the oriented z-flux. Neither weighted integral is a new object, and the caustic has been located before in the literature on harmonic trinomials; what is new is the closed evaluation of the multiplicity-weighted hierarchy across it.
Related
- Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments
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- Large Deviations for Controlled Branching Processes: Random Slopes, Harmonic Moments, and Transfer Principles
- Weighted moments and sharp convergence rates for supercritical multitype branching processes
- Some results on weighted p-harmonic ℓ-forms on weighted manifolds
BibTeX
@misc{bodis2026multiplicityweighted,
title = {{Multiplicity-Weighted Moments Across a Harmonic Caustic: Closed Forms with Order-Independent Breakpoints}},
author = {Tamás Bódis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.28370},
eprint = {2609.28370},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28370},
}
RIS
TY - UNPB
TI - Multiplicity-Weighted Moments Across a Harmonic Caustic: Closed Forms with Order-Independent Breakpoints
AU - Tamás Bódis
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.28370
UR - https://arxiv.org/abs/2609.28370
AB - While the Jacobian of a planar harmonic mapping keeps a constant sign, the oriented and the multiplicity-weighted surface elements agree up to that sign, and the two natural moment hierarchies coincide. The nearest classification results in the literature are stated under exactly this hypothesis; this paper works past it. Once the Jacobian changes sign, the critical circle acquires a caustic as its image, part of the plane is covered several times with opposite orientations, and the hierarchies part company: integrating a radial weight against the oriented surface element n_z lets overlapping sheets cancel, giving the degree-weighted moments J_n, while integrating against |n_z| lets every sheet count positively, giving the multiplicity-weighted moments J_n^±. These are the classical degree and area formulas for the same map, the second weighted by the multiplicity function (Banach indicatrix). It is the second family, the one the caustic threatens, for which a closed form is no longer automatic. Our main result is that it has one anyway: J_n^± is the total variation of the very same explicit primitive whose endpoint difference gives J_n. Crossing the caustic therefore costs only finitely many extra evaluation points, and they are the same points for every order n; for a monotone profile there is at most one interior breakpoint, at the critical radius itself. The family carrying this is the two-monomial planar harmonic mapping F(w)=w+iCw̅^k, w=p(u)e^iv, arising as the planar projection of a Fourier-perturbed surface. On the degree-weighted side J_n has a boundary-only closed form for every n≥ 0, the zeroth being the oriented z-flux. Neither weighted integral is a new object, and the caustic has been located before in the literature on harmonic trinomials; what is new is the closed evaluation of the multiplicity-weighted hierarchy across it.
ER -