We give an example of a function $f$ non-vanishing in the closed bidisk and the affine polynomial minimizing the norm of $1-pf$ in the Hardy space of the bidisk among all affine polynomials $p$. We show that this polynomial vanishes inside the bidisk. This provides a counterexample to the weakest form of a conjecture due to Shanks that has been open since 1980, with applications that arose from digital filter design. This counterexample has a simple form and follows naturally from previous work, where the phenomenon of zeros seeping into the unit disk was already observed for similar minimization problems in one variable.

For a distant observer with finite lifetime the main characteristic a black hole is trapping of light. Semiclassical description of black holes and especially the logical basis for construction of exotic horizonless models are based on two common but usually implicit assumptions. The first is a consequence of the cosmic censorship conjecture, namely that curvature scalars are finite at
apparent horizons. The second is that horizons form in finite asymptotic time (i.e. according to distant observers), a property implicitly assumed in conventional descriptions of black hole formation and evaporation. On the other hand, traversable wormholes are required to form in finite time and to be sufficiently regular by their design specifications.

Taking these as the only requirements within the semiclassical framework, one finds that in spherical symmetry only two classes of black/white hole solutions are admissible: each describing only evaporating black holes and expanding white holes. I review their properties and present the implications. For example, the null energy condition is violated in the vicinity of the outer and satisfied in the vicinity of the inner apparent/anti-trapping horizon. A test particle falls into a black hole in a finite time (according to a distant clock), and it is possible to be swallowed by a white hole. Kerr-Vaidya black holes share these qualitative features.

I conclude by discussing how the recent observation suggest that black holes are horizonless objects, and why some potential models of such objects, like wormholes, are ruled out.