Suppose I have a vector field $u$ defined on a domain $\Omega$ in $\mathbb{R}^2$, where the field is polynomial of some degree $K \geq 1$. That is, there exist polynomial basis functions $\Psi_i$ (which are real-valued), and nodal values $v_i$ as complex numbers, so that the field at every point is: $$ v(p \in \Omega) = \sum_i{v_i\psi_i(p)} $$ I am looking for an analytic way (if existing) to:
- Get the singularities of the fields. i.e., the locations $p_s$ where $v(p_s)=0$. This might not be possible for large $K$, but perhaps it has a nice closed form for $K \leq 2$?
- Given any point, determine the index of the field at that point. I should note that for linear fields ($K=1$) one can probably simply look at the eigenvalues of the Jacobian of the field, but I am unfamiliar with a way to get such indices for a higher $K$. Is there a rule that says that the possible (absolute) index is $K$ or some $O(K)$? it seems plausible from the example I've looked at.