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from the book "An Invitation to Algebraic Geometry"

Graphs of transcendental functions are not algebraic varieties. For example, the zero set of the function $y-e^x$ is not an algebraic variety.

I couldn't prove the statement in general .

I couldn't even prove the special case $y-e^x$ ...

I read this answer but i don't know how to deduce a contradiction.

Arsh Gh
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  • How about intersecting it with the line $y=1$? – Ahr May 20 '17 at 15:29
  • @A.Rod yes the answer that I linked up here exactly said that . but how does it work? – Arsh Gh May 20 '17 at 15:31
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    @ArshGh: That intersection is infinite and discrete, hence not algebraic. – Andrew D. Hwang May 20 '17 at 15:32
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    Oh, i should've clicked before answering then! =) If you intersect your graph with this line you should get an algebraic subset of the line i.e the emply set, the whole line, or a finite number of points. – Ahr May 20 '17 at 15:33
  • @A.Rod thank you . I'm new to algebraic geometry so i'm sorry if my question is trivial but, why an algebraic subset of the line should be like as you mentioned above? and why he Zariski topology on any plane curve is the cofinite topology?? – Arsh Gh May 20 '17 at 15:55
  • There no such thing as a trivial question. A subset of the line is closed iff it is the zero locus of a polynomial. To be a bit more precise, take $ax+by=0$ any line in $\mathbb{A^2}$, where you may assume that $a\neq 0$ (up to switching $a$ and $b$ in the following). A closed subset of the line is given by an equation $g(x,y)=0$ (or several but it does not matter) so you need to solve $g(-by/a,y)=0$ which is a polynomial in $y$ and as such has either fintiely many solutions, or vanish identically. – Ahr May 20 '17 at 16:04
  • For a general curve, instead of a line you may mimick that reasonning using for instance that $k[X,Y]$ is a UFD and maybe a bit of Nullstellensatz. – Ahr May 20 '17 at 16:05
  • @A.Rod thanks a lot. – Arsh Gh May 20 '17 at 16:09
  • @A.Rod sorry to bother you again. but for a beginner like me , is the book "An Invitation to Algebraic Geometry" a proper choice? – Arsh Gh May 20 '17 at 16:13
  • I wouldn't know, I have not read it. – Ahr May 20 '17 at 16:16
  • Let us continue this discussion in chat. – Ahr May 20 '17 at 16:52

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