# Resultant of multivariate polynomials

I know little about resultant. As for as I know, if two univariate polynomials $$f,g$$ have a common zero iff $$Res(f,g)=0$$, is it right?

Now I see an example using resultant of two multivariate polynomials to find the zero set these polynomials. The details are as follows:

In $$\mathbb{C}[y_1, y_2]$$, $$p_1=y_1^2+2y_1y_2-6y_1+y_2^2-6y_2+9,$$ $$p_2= y_1^2+2y_1y_2-6y_1+2y_2^2-9y_2+11.$$ To find solutions of $$p_1=0$$ and $$p_2=0$$. We calculate the resultant of $$p_1, p_2$$ with respect to $$y_1$$(see $$y_2$$ as common variable) and $$y_2$$ respectively: $$Res_{y_2}(p_1,p_2)=(y_1-1)^2(y_1-2)^2,$$ $$Res_{y_1}(p_1,p_2)=(y_2-1)^2(y_2-2)^2.$$ Plugging the four possible choices $$(1,1), (1,2), (2,1), (2,2)$$, we get the solution set $$\{ (1,2), (2,1)\}$$.

I am not familiar with resultant of multivariate polynomials, and I have two quesions here:

1. Why the solution set of $$p_1=0$$ and $$p_2=0$$ should have only this four possible choices?
2. If there are more variables $$y_i$$ and more $$p_i$$, is this also an efficient way to find the solution set of $$p_i=0$$?