Transvectant

Definition

If Q1,...,Q**n are functions of n variables x = (x1,...,x**n) and r ≥ 0 is an integer then the rth transvectant of these functions is a function of n variables given by \operatorname{Tr} \Omega^r(Q_1\otimes\cdots \otimes Q_n)where \Omega = \begin{vmatrix} \frac{\partial}{\partial x_{11}} & \cdots &\frac{\partial}{\partial x_{1n}} \ \vdots& \ddots & \vdots\ \frac{\partial}{\partial x_{n1}} & \cdots &\frac{\partial}{\partial x_{nn}} \end{vmatrix} is Cayley's Ω process, and the tensor product means take a product of functions with different variables x1,..., xn, and the trace operator Tr means setting all the vectors xk equal.

Examples

The zeroth transvectant is the product of the n functions. \operatorname{Tr} \Omega^0(Q_1\otimes\cdots \otimes Q_n) = \prod_k Q_kThe first transvectant is the Jacobian determinant of the n functions. \operatorname{Tr} \Omega^1(Q_1\otimes\cdots \otimes Q_n) = \det \begin{bmatrix} \partial_k Q_l \end{bmatrix}The second transvectant is a constant times the completely polarized form of the Hessian of the n functions.

When n = 2, the binary transvectants have an explicit formula:\operatorname{Tr} \Omega^k( f \otimes g ) = \sum_{l=0}^k (-1)^l \binom kl \partial_x^{k-l} \partial_y^l f \partial_y^{k-l} \partial_l^l gwhich can be more succinctly written asf \left(\overleftarrow{\partial_{x}} \cdot \overrightarrow{\partial_{y}}-\overleftarrow{\partial_{y}} \cdot \overrightarrow{\partial_{x}}\right)^k gwhere the arrows denote the function to be taken the derivative of. This notation is used in Moyal product.

Applications

References