Transvectant

Transvectant is a construction from classical invariant theory that produces new covariants from two binary forms by a bilinear differential operation. If f and g are homogeneous polynomials in two variables x and y of degrees m and n, and k is an integer with 0 ≤ k ≤ min(m,n), then the k-th transvectant, denoted (f,g)^k, is a homogeneous form in x and y of degree m+n−2k. The transvectant is linear in each argument and behaves compatibly with the action of the special linear group SL(2), so it is used to build invariants and covariants of binary forms.

A convenient explicit formula for the k-th transvectant is

(f,g)^k = (1/k!) ∑_{i=0}^k (-1)^i binom(k,i) ∂^{k}f/∂x^{k−i} ∂y^{i} · ∂^{k}g/∂x^{i} ∂y^{k−i},

where the derivatives are taken with respect to x and y and the result is a homogeneous polynomial in x and y of degree m+n−2k. Different conventions exist, but all definitions share the same basic idea: kth transvection combines f and g by a complete antisymmetrized pairing of derivatives.

Special cases illustrate the range of the construction. When k = 0, (f,g)^0 = fg is the ordinary product. When f and g are linear forms, the case k = 1 yields the determinant formed from their coefficients, an invariant of SL(2). In general, (f,g)^k is a covariant; if m+n−2k = 0, it becomes an invariant scalar.

Transvectants were developed by Cayley and Sylvester and remain a foundational tool for constructing the algebra of covariants and invariants of binary forms, as well as for expressing discriminants and related classical invariants.