# Intersection of Plane Curve and Line

Question: Suppose $$C$$ is an affine plane curve and L is a line in $$A^2(K)$$, $$L\not\subset C$$. Suppose $$C=V(F)$$ and $$F\in K[x,y]$$ a polynomial of degree n. Show that $$L\cap C$$ is a finite set of no more than n points.

My attemp: Btw, I saw this question nearby similar question but I can't understand writings about the proof.

First, I don't know meaning of affine plane curve. Now, $$F$$ is polynomial so $$V(F)$$ is zero set of polynomial and $$L$$ is a line in $$A^2$$ so it is something like $$y=ax+b$$. If a point $$(a,b)$$ in $$L \cap C$$ then it satisfies the following $$F(a,b)=0$$ and we have this line $$y=ax+b$$.

I am stuck with it now.

• An affine plane has the form $a_0^{\intercal} \, x + b_0 = 0$ and a line is a $n-1$ dimensional plane also with form $$a_1^{\intercal} \, x + b_1 = 0$$. I do not understand the notation $A^2$, $V(F)$ and $F \in K[x, y]$. Are there additional information? Apr 19 at 18:09
• Hint: Consider $F(x, ax+b)$. This is a polynomial in one variable of degree at most $n$. Also, note that a vertical line $x = c$ can't be written in the form you've given, so some further argument is needed. Apr 19 at 18:14
• $V(F)$ is vanishing set , F is polynomial in $K[x,y]$ and $A^2 = \{ (a,b) : a,b \in K \}$ and $K$ is field. Apr 19 at 18:15
• @BrunoHenriquePeixoto these are well-known terms from an area of math called algebraic geometry. What you're doing right now is sort of the equivalent of wandering in to a calculus question for the first time and asking what $\int$ means - sure, it would be good for you to get an answer for that, but this isn't really the right place for those questions. Apr 19 at 18:30