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 $p_i=0$?

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