Parabolic function

Definition

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