  We introduce in this paper an equivalence notion for submersions $U \to \R$,
$U$ open in $\R^2$, which makes it possible to identify a smooth planar curve
with a unique class of submersions. This idea, which extends to the nonlinear
setting the construction of a dual projective space, provides a systematic way
to handle global implicit descriptions of smooth planar curves. We then apply
this framework to model nonlinear electrical devices as {\em classes of
equivalent functions}. In this setting, linearization naturally accommodates
incremental resistances (and other analogous notions) in homogeneous terms.
This approach, combined with a projectively-weighted version of the matrix-tree
theorem, makes it possible to formulate and address in great generality several
problems in nonlinear circuit theory. In particular, we tackle unique
solvability problems in resistive circuits, and discuss a general expression
for the characteristic polynomial of dynamic circuits at equilibria. Previously
known results, which were derived in the literature under unnecessarily
restrictive working assumptions, are simply obtained here by using
dehomogenization. Our results are shown to apply also to circuits with
memristors. We finally present a detailed, graph-theoretic study of certain
stationary bifurcations in nonlinear circuits using the formalism here
introduced.
