On the Wronskian combinants of binary forms

Given a sequence A=(A1,…,Ar)A=(A1,…,Ar) of binary dd-ics, we construct a set of combinants C={Cq:0≤q≤r,q≠1}C={Cq:0≤q≤r,q≠1}, to be called the Wronskian combinants   of AA. We show that the span of AA can be recovered from CC as the solution space of an SL(2)SL(2)-invariant differential equation. The Wronskian combinants define a projective imbedding of the Grassmannian G(r,Sd)G(r,Sd), and, as a corollary, any other combinant of AA is expressible as a compound transvectant in CC.Our main result characterises those sequences of binary forms that can arise as Wronskian combinants; namely, they are the ones such that the associated differential equation has the maximal number of linearly independent polynomial solutions. Along the way we deduce some identities which relate Wronskians to transvectants. We also calculate compound transvectant formulae for CC in the case r=3r=3.