  The anomalous bulk-boundary correspondence in non-Hermitian systems featuring
an intricate interplay between skin and boundary modes has attracted enormous
theoretical and experimental attention. Still, in dimensions higher than one,
this interplay remains much less understood. Here we provide insights from
exact analytical solutions of a large class of models in any dimension, $d$,
with open boundaries in $d_c \le d$ directions and by tracking their
topological origin. Specifically, we show that Amoeba theory accounting for the
separation gaps of the bulk skin modes augmented with higher-dimensional
generalizations of the biorthogonal polarization and the generalized Brillouin
zone approaches accounting for boundary modes provide a comprehensive
understanding of these systems.
