  Let R be a commutative ring with 1. We prove that every homogeneous
polynomial f(x_0,x_1,x_2) in R[x_0,x_1,x_2] up to degree 5 admits a linear
Pfaffian R-representation. We believe that conceptually we give the shortest
self-contained proof possible: we exhibit explicitly such a representation. In
this sense, we generalize (up to degree 5) a result due to A. Beauville about
the existence of Pfaffian representations for any smooth plane curve of any
degree.
