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I just started to read Algebraic Geometry and when I was going through exercises in Hartshorne's algebraic geometry book I came about the following question.

Consider the plane curve $y = x^2$. Then $Y = \mathcal{Z}(x^2-y)$ is the set values where this polynomial vanishes. As the polynomial $x^2-y$ is irreducible in $k[x,y]$ (where $k$ is algebraically closed field) so $Y$ will be an affine algebraic variety as its ideal will be a prime ideal in $k[x,y]$. My question is:

If we consider the $\textbf{graph}$ of the function $y = x^2$ in $\mathbb{R^2}$ and the $\textbf{affine variety}$ $Y$ which is basically the set of zeros of the polynomial $x^2-y$, then:

1) What is the difference between them? Does the motivation of affine varities come from the graph if we are considering only a single curve and the corresponding polynomial in $k[x, y]$ is irreducible?

2) How does the above situation apply to a finite number of curves in the sense if we plot them in $\mathbb{R^2}$ and take the set of common intersection points of all the curves, how that set is going to be different from the algebraic set that is generated by the corresponding polynomial equations of these curves in $k[x,y]$?

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  • $\begingroup$ If the graph is given by a polynomial, they are the same thing. $\endgroup$ – user40276 Sep 3 '14 at 15:35
  • $\begingroup$ $\mathbb R$ is not algebraically closed. You can think of the graph as the "real part" of $Z(y - x^2) \subset \mathbb C^2$. $\endgroup$ – Ayman Hourieh Sep 3 '14 at 16:45
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Suppose $ y-x^2 \in k[x,y]$, Where $k$ is algebraic closed, $Z(y-x^2)$ is called an affine curve and it is an affine Variety. affine varieties used as a tool for studying algebraic constructions, and vice versa. If $V_1=Z(f_1) , \cdots , V_n=Z(f_n)$ are affine curves, then $V_1 \cap \cdots \cap V_n = Z((f_1) \cup \cdots \cup (f_n)) $.

There is a branch of algebraic geometry, real algebraic geometry, for studying real algebraic sets i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings).

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