On the invariant theory of the Bezoutiant

Abstract

We study the classical invariant theory of the Bezoutiant R(A,B) of a pair of binary forms A,B. It is shown that R(A,B) admits a Taylor expansion whose coefficients are (essentially) the odd transvectants (A,B)_{2r+1}. Moreover, R(A,B) is entirely determined by the first two terms M = (A,B)_1, N =(A,B)_3. Using the Plucker relations, we give equivariant formulae which express the higher transvectants (A,B)_5, (A,B)_7 in terms of M,N. We also describe a `reduction formula' which calculates B from the knowledge of A and R(A,B).