I want to describe all polynomials that have the basis $x_1^{d_1}x_2^{d_2}...x_n^{d_n}$ where $\sum_i d_i = k$, $d_i\geq 0$, $d_i\in \mathbb{Z}$. I want to encode this in a nice way.
For example, if $n=2$, each basis vector can be written $x_1^{d}x_2^{k-d}$ and can be encoded in a $(k+1)$-component vector $x$ quite nicely where $x^d$ is the coefficient for $x_1^dx_2^{k-d}$. Any linear operations on such a polynomial can then be done with 2D matrices.
If $n=3$, the polynomial is $x_1^{d_1}x_2^{d_2}x_3^{k-d_1-d_2}$ which can only be encoded nicely in a $(k+1)\times(k+1)$ matrix $X$ where $X^{d_1}_{d_2}$ is the coefficient of $x_1^{d_1}x_2^{d_2}x_3^{k-d_1-d_2}$. Any operations on this would then need "3D matrices" to operate naturally on and I am really quite unaware on what the theory on this is.
I need to work for general cases of $n$ so this is very important. Is this what tensor calculus is about or is the math slightly different?
I basically need to find the ranks of some of these higher dimensional matrices which is really quite unintuitive for me as I can't visualize them. Any help is appreciated!