  In this paper we take the first step towards a non-diagrammatic formulation
of the Pinch Technique. In particular we proceed into a systematic
identification of the parts of the one-loop and two-loop Feynman diagrams that
are exchanged during the pinching process in terms of unphysical ghost Green's
functions; the latter appear in the standard Slavnov-Taylor identity satisfied
by the tree-level and one-loop three-gluon vertex. This identification allows
for the consistent generalization of the intrinsic pinch technique to two
loops, through the collective treatment of entire sets of diagrams, instead of
the laborious algebraic manipulation of individual graphs, and sets up the
stage for the generalization of the method to all orders. We show that the task
of comparing the effective Green's functions obtained by the Pinch Technique
with those computed in the background field method Feynman gauge is
significantly facilitated when employing the powerful quantization framework of
Batalin and Vilkovisky. This formalism allows for the derivation of a set of
useful non-linear identities, which express the Background Field Method Green's
functions in terms of the conventional (quantum) ones and auxiliary Green's
functions involving the background source and the gluonic anti-field; these
latter Green's functions are subsequently related by means of a Schwinger-Dyson
type of equation to the ghost Green's functions appearing in the aforementioned
Slavnov-Taylor identity.
