Determinant Characteristics and Argument-Principle Certification for Visible Poles in Meromorphic Continuation

Abstract

We study outward meromorphic continuation from circular boundary data in the unit disk. The unknown function is holomorphic in the unit disk and admits a meromorphic continuation to a larger disk, where finitely many exterior simple poles are superposed on an unknown holomorphic background. The positive Fourier coefficients of the boundary trace are Taylor coefficients at the origin, and exterior poles generate a finite exponential-sum component in these coefficients. We introduce shifted determinant characteristics and prove that, in the pure finite-pole model, the determinant for the correct order factors exactly into a nonzero constant times the polynomial whose zeros are the reciprocals of the exterior poles. The same zero set is obtained for noiseless equispaced discrete Fourier coefficients; sampling changes only the amplitudes through an aliasing factor. For data containing a holomorphic background, discretization effects, and noise, roots of a single empirical determinant are only candidate reciprocal poles. We therefore propose a root-propose and contour-certify procedure: determinant roots generate candidate regions, while local argument-principle counts, contour moments, empirical margins, and persistence over determinant orders and shifts certify visible poles. A Rouch\'e-type perturbation analysis gives sufficient conditions for stable local zero counts and explains how residues, pole separation, distance to the target annulus boundary, shifts, and noise affect visibility. Numerical experiments verify the pure-pole identity, demonstrate certification under background and noise, and show that high noise, weak residues, boundary-near poles, and close poles naturally lead to partial recovery of contour-certified visible poles.