The signature of a membrane is a sequence of tensors whose entries are
iterated integrals. We study algebraic properties of membrane signatures,
with a focus on signature matrices of polynomial and piecewise bilinear
membranes. Generalizing known results for path signatures, we show that
the two families of membranes admit the same set of signature matrices
and we examine the corresponding affine varieties. In particular, we prove
that there are no algebraic relations on signature matrices of membranes,
in contrast to the path case. We complement our results by a linear time
algorithm for the computation of signature tensors for piecewise bilinear
membranes.
Keywords
two-parameter signatures, affine algebraic varieties,
matrix congruence, geometry of paths and membranes,
iterated integrals
