A bivariate polynomial of degree $m+n$ is,

$ p(x,y) = \sum_{k=1}^n\sum_{j=1}^m a_{jk}x^ky^j$

where $a_{mn}\neq0$ and $a_{jk}\in\mathbb{R}$ for $1\leq j\leq m$, $1\leq k\leq n$.

I would like to understand how the roots of a bivariate polynomial behave. It is clear that the roots cannot form patches (unless $p$ is the zero polynomial) but I'm sure about the following:

If some of the roots form a curve, what properties does this curve have? Can it bifurcate? Can the curve be parameterised to a univariate polynomial? Can the curve have end points (which are not $\pm\infty$)?

What is the maximum number of isolated roots of $p(x,y)$? What is the maximum number of zero curves?

How badly ill-conditioned is bivariate polynomial root finding? For example the polynomial $p(x,y) = x^2+2x+1$ has one zero curve but $p(x,y) = x^2 + (2-\epsilon)x+1$ has two zero curves. Is there a polynomial $p(x,y)$ with a zero curve but with a small perturbation of the coefficients has only isolated zeros?

I guess the fundamental theorem of algebra does not hold. The polynomial $p(x,y) = x^n-y$ seems to be a counterexample. Is there a multivariate function theorem?

This post is a barrage of questions, but I feel they are all intrinsically related and a person who can answer one of them is likely to be able to give answers to them all.

Thank you.