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Given a polynomial ODE in $n$-dimensions of maximal degree $d$ $$ \dot{x}_j=f_j(x)=\sum_{i_1,\dots,i_n=1}^d a_{i_1,\dots,i_n}^j x_1^{i_1}\dots x_n^{i_n} \quad \forall j=1,\ldots,n $$ we define ...

Let's assume we are given a degree $d$ polynomial VF as a map $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ $$f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a^j_{i_{1},\dots,i_{n}}x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$$...