  Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field
of characteristic zero, we focus on the first syzygy module Z of the transposed
Jacobian module D(f), whose elements are called differential syzygies of f.
There is a distinct submodule P of Z coming from the polynomial relations of f
through its transposed Jacobian matrix, the elements of which are called polar
syzygies of f. We say that f is polarizable if equality P=Z holds. This paper
is concerned with the situation where f are monomials of degree 2, in which
case one can naturally associate to them a graph G(f) with loops and translate
the problem into a combinatorial one. A main result is a complete combinatorial
characterization of polarizability in terms of special configurations in this
graph. As a consequence, we show that polarizability implies normality of the
subalgebra k[f] of R and that the converse holds provided the graph G(f) is
free of certain degenerate configurations. One main combinatorial class of
polarizability is the class of polymatroidal sets. We also prove that if the
edge graph of G(f) has diameter at most 2 then f is polarizable. We establish a
curious connection with birationality of rational maps defined by monomial
quadrics.
