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: 


1. 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 $\pm\infty$)?
2. What is the maximum number of isolated roots of $p(x,y)$? What is the maximum number of zero curves?
3. 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?
4. 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. 



Roots of bivariate polynomials
I don't think most of your questions are appropriate for MO, you should try math.stackexchange.com if you have any follow-up questions, as it sounds as if your knowledge and questions are at advanced undergraduate/beginning graduate level, but I'll try to answer some of them. A good reference is Fulton "Algebraic Curves" or perhaps some older books, like Walker's which go deeper into real points. 


First a degree $d=m+n$ polynomial is usually $\sum\_{j+k \le d} a\_{jk}x^jy^k$, which is a little more general than what you wrote.


1. The zero set is a union of finitely many (real smooth) curves and points and has a finite number of singularities (nodes, cusps and higher) which look like multiple crosses and pointy bits (you have to look at a picture). It cannot end abruptly and, if by bifurcation you mean one curve separate into two, then I don't think you can have that. The curve can be parametrized by univariate polynomials only in very special cases (irreducible, genus zero and only one point at infinity).
2. At most $d^2$ isolated points (can improve bound slightly) because they are singular. There is also at most some quadratic function of $d$ number of "zero curves", I forget the exact formula. This is known as Harnack's theorem.
3. I don't know, ask a numerical analyst. Edit: Re last part of question. Having a curve of solutions is an open condition on the coefficients. An useful example is $x^2+y^2=a$, and look at $a$ small, positive, negative or zero.
4. Question doesn't make sense.


