Signature matrices of membranes

We prove that, unlike in the case of paths, the signature matrix of a membrane does not satisfy any algebraic relations. We derive novel closed-form expressions for the signatures of polynomial membranes and piecewise bilinear interpolations for arbitrary $2$-parameter data in $d$-dimensional space. We show that these two families of membranes admit the same set of signature matrices and scrutinize the corresponding affine variety.