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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.

Thanks in advance

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  • $\begingroup$ 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? $\endgroup$ Apr 19 at 18:09
  • $\begingroup$ mathworld.wolfram.com/AffineFunction.html $\endgroup$ Apr 19 at 18:10
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    $\begingroup$ 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. $\endgroup$ Apr 19 at 18:14
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    $\begingroup$ $V(F)$ is vanishing set , F is polynomial in $K[x,y]$ and $A^2 = \{ (a,b) : a,b \in K \}$ and $K$ is field. $\endgroup$
    – nozalp10
    Apr 19 at 18:15
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    $\begingroup$ @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. $\endgroup$
    – KReiser
    Apr 19 at 18:30

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