  The non-Hermitian skin effect, characterized by a proliferation of
exponentially-localized edge modes, has led to numerous novel physical
phenomena that challenge the limits of conventional band theory. In sharp
contrast to the traditional exponential localization, this manuscript reports a
new kind of non-Hermitian skin effect, which we term the ``algebraic
non-Hermitian skin effect." This effect emerges across a diverse spectrum of
non-Hermitian systems in both two- and higher space dimensions. For 2D systems
with algebraic non-Hermitian skin effect, on geometries such as a torus or
cylinder, these systems exhibit behavior reminiscent of the conventional
non-Hermitian skin effect, where eigenmodes are either bulk Bloch waves (on a
torus) or exponentially localized edge modes (on a cylinder). However, if the
same system is placed on a disk or any geometrical shape featuring open
boundaries in all directions, the skin modes immediately transform into the
algebraic form, with amplitude decaying as a power-law function of the distance
from the boundary. To explore these novel effects, we formulate a unified
generalized Brillouin zone (GBZ) framework that is universally applicable to
all variations of non-Hermitian skin effects across any spatial dimension,
developed through the usage of a generalized transfer-matrix approach. We find
that in a $d$-dimensional non-Hermitian system, in general, the GBZ manifold's
dimensionality must fall into the range from $d$ to $2d-1$, denoted by ${d \leq
\dim\text{GBZ} \leq 2d-1}$. In 1D, this inequality is trivial because the upper
and lower bounds converge, forcing the GBZ's dimensionality to match with that
of the physical space. However, in 2D and above, this inequality indicates that
there is no obligation for the GBZ's dimensionality to concur with the physical
space's dimensionality, which gives rise to a new class of non-Hermitian skin
effects.
