Contour-count indicator fields for visible pole clusters in meromorphic continuation
arXiv:2607.12256
The paper introduces a contour-count indicator method that uses determinant characteristics from Fourier coefficients to image and certify clusters of visible poles in outward meromorphic continuation, with post‑processing via persistent homology for robust detection.
Abstract
We develop a contour-count indicator method for visible pole clusters in outward meromorphic continuation from circular boundary data. The method starts from determinant characteristics built from positive Fourier coefficients. In the pure finite-pole model, the correct determinant characteristic factors into a polynomial whose zeros are the reciprocals of the exterior poles. In the presence of a holomorphic background, finite sampling, and noise, roots of individual determinants are unstable and are used only as local evidence. We aggregate this evidence into a scalar indicator field on the reciprocal pole plane: at each sampling point, the field records the fraction of determinant orders and shifts for which a small contour centered at that point encloses exactly one empirical determinant zero. The resulting field plays the role of a sampling-type imaging functional for pole visibility. Fixed superlevel sets give visible-pole clusters, while zero-dimensional persistent homology is used only as a threshold-robust post-processing step. We prove deterministic results linking pure-pole contour counts, Rouché stability, indicator-field contrast, fixed-threshold component recovery, and persistence-gap stability. These results explain why isolated poles with sufficient residue and separation generate stable high-value components, whereas weak, close, boundary-near, or noise-dominated poles may give low, short-lived, or merged components. The framework is a cluster-certification and imaging method, not an unconditional all-pole recovery procedure.