On the Wronskian combinants of binary forms |
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Authors: | Abdelmalek Abdesselam Jaydeep Chipalkatti |
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Affiliation: | a Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada b LAGA, Institut Galilée, CNRS UMR 7539, Université Paris XIII, 99 Avenue J.B. Clément, F93430 Villetaneuse, France c Department of Mathematics, University of Manitoba, 433 Machray Hall, Winnipeg MB R3T 2N2, Canada |
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Abstract: | Given a sequence A=(A1,…,Ar) of binary d-ics, we construct a set of combinants C={Cq:0≤q≤r,q≠1}, to be called the Wronskian combinants of A. We show that the span of A can be recovered from C as the solution space of an SL(2)-invariant differential equation. The Wronskian combinants define a projective imbedding of the Grassmannian G(r,Sd), and, as a corollary, any other combinant of A is expressible as a compound transvectant in C.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 C in the case r=3. |
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Keywords: | 13A50 |
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