Suppose we have a polynomial function $f:\mathbb{C}^n\to\mathbb{C}$, it can be written as $f(X_1,\ldots\,X_N) = \sum_{j=1}^m c_j X_1^{a_{j_1}}\cdot\ldots\cdot X_n^{a_{j_n}}$. We may want to compute the $k$-th derivative of $f$ in some point $z\in\mathbb{C}^n$, which is a $k$-linear function $D^kf(z):\underbrace{\mathbb{C}^n\times\mathbb{C}^n}_{k \text{ times}}\to\mathbb{C}$.
To be more precise, this function is given by $$D^kf(z)(v_1,\ldots,v_k) = \sum_{i_1,\ldots,i_k=1}^n v_{1,i_1}\cdot\ldots\cdot v_{k,i_k} \frac{\partial^k f(z)}{\partial X_{i_1}\ldots\partial X_{i_k}}, $$ where $v_j = (v_{j,1},v_{j,2},\ldots,v_{j,n})$ for $j=1\ldots k$.
Note that we can write $$D^kf(z) = \sum_{i_1,\ldots,i_k=1}^n dx_{i_1}\otimes\ldots\otimes dx_{i_k}\frac{\partial^k f(z)}{\partial X_{i_1}\ldots\partial X_{i_k}},$$ so I suspect Mathematica have a built in function for this, because it works with tensors (although the documentation wasn't very helpful for me in this case).
I have two questions:
1) How can I make Mathematica to compute $D^kf(z)(v_1,\ldots,v_k)$ when I give only the function $f$, the point $z\in\mathbb{C}^n$ and the vectors $v_1,\ldots v_k\in\mathbb{C}^n$ as input?
2) If my first question has a positive answer, I want to make Mathematica to compute $$\|D^kf(z)\| = \max_{v1,\ldots,v_k\in\mathbb{C}^n, \|v_j\|=1} |D^kf(z)|.$$
How can I do this?
Thank you very much for your help.
PS: the fact that $f$ is a polynomial is not so important, what is really important is that $f$ is differentiable $k$ times. Also, the mentioned norms $\|v_j\|$ are the usual norms $\|v_j\|=\sqrt{|v_{j,1}|^2+\ldots+|v_{j,n}|^2}$ (note that each $v_{j,l}$ is a complex number, so it is necessary to compute the absolute value).
