Parabolic function

Definition

Let $(m_1,m_2,\dots ,m_r)$ 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 $(m_1,m_2,\dots ,m_r)$ if for any $k\le r$, 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 $(m_{!},m_2,\dots ,m_r)$.