I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) is an «algebraic variety» in the sense that it coincides with the intersection of vanishing locii of polynomials in the finite jets of the $f_j$'s. In fact this derives from the same result for formal power series $\mathbb C[[z_1,…,z_m]]$.

This kind of result has been extensively studied for polynomials, leading to Sylvester and Macaulay matrices (my proof involves a sequence of matrices containing those for the part of least homogeneous degree). Since I'm no specialist of elimination theory I wasn't able to find any reference regarding this topic for formal power series (over a field of characteristic $0$), although I'd be quite surprised indeed to learn that this fact is not known.

I'd be thankful to anybody pointing me a name or reference regarding this matter.