# Fundamental Theorem of Algebra for Two Variables

Is there an extension for the Fundamental Theorem of Algebra for Two or more variables, such in case of polynomials systems:

$\begin{cases} f(x, y) = 0 \\ g(x, y) = 0 \end{cases}$

For single-variable polynomials, the Theorem states that nth-degree polynomials implies n complex roots. And about two or more variables, there is such extension? Perhaps a sum of degrees or the max number of degrees of the variables?

• No there isn't. You can't make such a guarantee. However, if you set any one of the variables to be constant, then again you can apply the theorem. Sep 7, 2018 at 18:31

An example that shows why you need the projective plane is $f(z,w)=w$, $g(z,w)=w-1$. Then $\{f=0\}$ and $\{g=0\}$ are parallel complex "lines" that do not intersect in $\mathbb{C}^2$, but the projective plane has "points at infinity" where parallel lines intersect.
• (I am not an algebraic geometer, so I won't give you the formal definition). Basically you count a point once if the intersection of $N_f=\{f(x,y)=0\}$ and $N_g=\{g(x,y)=0\}$ at that point is transversal, i.e. if the tangents to $N_f$ and $N_g$ span $\mathbb{C}^2$. If the tangents are the same the intersection number is at least $2$. If higher order approximations are the same, you have higher intersection number. Some nice examples are in the Wikipedia article I linked. Sep 7, 2018 at 19:45