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. 



Continuity in the roots of an algebraic variety with respect to the coordinates
Of course, (real) solutions can just disappear under small perturbations, so you'd definitely have to assume that the number of solutions, counted appropriately, stays constant. But that's not enough either because you can move zeros around between factors. Consider for example
$$
x=a, \quad (x^2+y^2)(xy-1)=0 .
$$
For $a=0$, this is solved by $(0,0)$, but for small $a\not=0$, we obtain $(a,1/a)$ as the unique solution.


