Parabolic function

Definition

Let $(m_{1}, m_{2}, \dots, m_{r})$ be a tuple of integers whose sum is $n$ . Then, a function $f : R^{n} \to R^{n}$ is said to be of parabolic type $(m_{1}, m_{2}, \dots, m_{r})$ if for any $k \leq r$ , the effect on the first $m_{1} + m_{2} \dots m_{k}$ coordinates is independent of the values of the remaining coordinates.

If $f$ is also continuously differentiable, this is equivalent to the condition that the Jacobian matrix of $f$ be of parabolic type $(m_{!}, m_{2}, \dots, m_{r})$ .