If I'm slightly misusing definitions forgive me I'm not an algebraist.
I have $N$ polynomials $f_n(x)$, $n=1,\ldots,N$ where $x\in \mathbb{R}^N$ and the set $\{x:f_n(x)=0\text{ for all }n\}$ is finite. If I slightly change the coordinates in the polynomials are there some conditions and a result that lets me ensure that the roots themselves change continuously?
As for the work I've done so far, I can apply the Implicit Function Theorem so that as long as the matrix $A_{ij}=\frac{d}{dx_j}f_i(x)$ is invertible at all roots I can ensure that the roots change continuously and there is no bifurcation. And in the complement of the neighbourhoods of these points, one of my polynomials is $x_1^2+\ldots+x_N^2=1$ so by compactness no other roots show up. But stitching these two results together is a pain and it seems likely that this is a standard result in a topic I'm not familiar with.