Let $C_f$ be the algebraic curve in $\mathbb{C^2}$ which is defined by the polynomial $f\in \mathbb{C}[X,Y] $ and $C_g$ be the algebraic curve in $\mathbb{C^2}$ which is defined by the polynomial $g\in \mathbb{C}[X,Y] $. We will use the symbols $V(f),\ V(g) $ for the zero sets of $C_f$, $C_g$ respectively. We want to show the following proposition:
Proposition If $C_f, \ C_g$ have not common components then $|V(f)\cap V(g)|<\infty$.
Proof
- First of all lets asume that the polynomials $f,g $ with coefficients in $\mathbb{C}[X]$, have positive degree and $R$ is the resultant of $f,g $ in the variable $Y$. Then, there are $A,B\in\mathbb{C}[X,Y]: R=Af+Bg$. So, if $(x,y)\in V(f)\cap V(g)$ then $R(x)=0 $ and that means that there are maximum finite values for $x$ and consequently for $y$.
- Now, lets asume that $f=a(X-ρ_1)\cdot...\cdot (X-ρ_s)$ and $g=b_0(X)+b_1(X)Y +...+ b_r(X)Y^r$. As $f,g$ have not common factor $\forall i,\exists j: b_j(ρ_i) \neq 0.$ If $j\geq1$ then there are finite values for $y$ such that $(ρ_i,y) \in V(f)\cap V(g).$ If $b_0(ρ_i) \neq 0$ and $b_1(ρ_i)=...=b_r(ρ_i)=0 \implies \not \exists y : (ρ_i,y) \in V(f)\cap V(g)$.
Question: I am studying this proof but I don't understand the second bullet. Could anybody explain me / expand the second bullet please?
Thank you.