Abstract.We reconsider the classical problem of representing a finite number of forms of degree d in the polynomial ring over n + 1 variables as scalar combinations of powers of linear forms. We define a geometric construct called a 'grove', which, in a number of cases, allows us to determine the dimension of the space of forms which can be so represented for a fixed number of summands. We also present two new examples, where this dimension turns out to be less than what a naïve parameter count would predict. Mathematics Subject Classification (2000). 14N15, 51N35.

No abstract

We revisit an old problem in classical invariant theory, viz. that of giving algebraic conditions for a binary form to have linear factors with assigned multiplicities. We construct a complex of SL 2 -representations such that the desired algebraic conditions are expressible as a specific cohomology group of this complex.

Given a sequence A = (A 1 , . . . , A r ) of binary d-ics, we construct a set of combinants C = {C q : 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 S L(2)-invariant differential equation. The Wronskian combinants define a projective imbedding of the Grassmannian G(r, S d ), 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.

Abstract. Given integers n, d, e with 1 ≤ e < d 2 , let X ⊆ P ( d+n d )−1 denote the locus of degree d hypersurfaces in P n which are supported on two hyperplanes with multiplicities d − e and e. Thus X is the BrillGordan locus associated to the partition (d − e, e). The main result of the paper is an exact determination of the Castelnuovo regularity of the ideal of X. Moreover we show that X is r-normal for r ≥ 3.In the case of binary forms (i.e., for n = 1) we give an invariant theoretic description of the ideal generators, and furthermore exhibit a set of two covariants which define this locus set-theoretically.In addition to the standard cohomological tools in algebraic geometry, the proof crucially relies on the nonvanishing of certain 3j-symbols from the quantum theory of angular momentum.