We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the polydisk possess non-tangential limits at every boundary point. We relate higher non-tangential regularity and distinguished boundary behavior of bounded rational functions to geometric properties of the zero sets of stable polynomials via our local descriptions. For a fixed stable polynomial $p$, we analyze the ideal of numerators $q$ such that $q/p$ is bounded on the bi-upper half plane. We completely characterize this ideal in several geometrically interesting situations including smooth points, double points, and ordinary multiple points of $p$. Finally, we analyze integrability properties of bounded rational functions and their derivatives on the bidisk.
A short and simple proof of necessity in the McCullough-Quiggin characterization of positive semi-definite kernels with the complete Pick property is presented.
Motivated by the discovery of large-scale ionized clouds around AGN host galaxies, and particularly the large fraction of those which are consistent with photoionized gaseous tidal debris, we have searched for [O III] emission over wide fields around a set of Seyfert galaxies previously mapped in H I, many of which show extended gas features. The detection threshold was designed to reach mean emission-line surface brightness 10 times fainter than seen in such AGN-ionized clouds as Hannys Voorwerp, so that similar structures at larger distances (and ages) could be detected. Of 24 Seyfert galaxies, we find one extended emission feature, a discrete cloud projected 12 kpc from the center of Mkn 1 and spanning a transverse extent of 8 kpc. Optical spectroscopy of the Mkn 1 cloud confirms its redshift association with the Mkn 1- NGC 451 galaxy pair, shows it to closely match the kinematics of nearby H I, and reveals emission-line ratios requiring photoionization by the AGN at roughly the direct observed luminosity of the nucleus. Given the small fraction of H I features with detected [O III] emission, we constrain the typical opening angle of ionization cones in Seyfert galaxies to be of order 20 deg, if active episodes are long compared to the light-travel times involved. An appendix presents a derivation of an analytical expression for the probability of intersection of a cone with randomly oriented arcs, approximating the geometry of H I clouds and tails exposed to ionization cones. For the entire sample, the full opening angle of bicones must be <20 deg if the AGN are continuously bright for scales longer than the light-travel times across the H I structures. Since many ionization cones are observed to be much broader than this, our low detection fraction may add to evidence for the ubiquity of strong variations in AGN luminosity on scales 10,000-100,000 years.
