  One fundamental question in the context of the geometric complexity theory
approach to the VP vs. VNP conjecture is whether VP = $\overline{\textrm{VP}}$,
where VP is the class of families of polynomials that are of polynomial degree
and can be computed by arithmetic circuits of polynomial size, and
$\overline{\textrm{VP}}$ is the class of families of polynomials that are of
polynomial degree and can be approximated infinitesimally closely by arithmetic
circuits of polynomial size. The goal of this article is to study the
conjecture in (Mulmuley, FOCS 2012) that $\overline{\textrm{VP}}$ is not
contained in VP.
  Towards that end, we introduce three degenerations of VP (i.e., sets of
points in $\overline{\textrm{VP}}$), namely the stable degeneration Stable-VP,
the Newton degeneration Newton-VP, and the p-definable one-parameter
degeneration VP*. We also introduce analogous degenerations of VNP. We show
that Stable-VP $\subseteq$ Newton-VP $\subseteq$ VP* $\subseteq$ VNP, and
Stable-VNP = Newton-VNP = VNP* = VNP. The three notions of degenerations and
the proof of this result shed light on the problem of separating
$\overline{\textrm{VP}}$ from VP.
  Although we do not yet construct explicit candidates for the polynomial
families in $\overline{\textrm{VP}}\setminus$VP, we prove results which tell us
where not to look for such families. Specifically, we demonstrate that the
families in Newton-VP $\setminus$ VP based on semi-invariants of quivers would
have to be non-generic by showing that, for many finite quivers (including some
wild ones), any Newton degeneration of a generic semi-invariant can be computed
by a circuit of polynomial size. We also show that the Newton degenerations of
perfect matching Pfaffians, monotone arithmetic circuits over the reals, and
Schur polynomials have polynomial-size circuits.
