Affine Algebraic Curves - Kunz - Definition 1.1

[Math Amateur]

I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves"

I need help with some aspects of Kunz' Definition 1.1.

The relevant text from Kunz' book is as follows:View attachment 4556In the above text, Kunz writes the following:

" ... ... If $$K_0 \subset K$$ is a subring and $$\Gamma = \mathscr{V} (f)$$ for a nonconstant polynomial $$f \in K_0 [X,Y]$$ ... ..."


My question is as follows:

Given $$f \in K_0 [X,Y]$$ means that the co-efficients of $$f$$ come from $$K_0$$ ... so if

$$f = aX + bY + c $$

then $$a,b,c$$ come from $$K_0$$ ... that is $$a,b,c \in K_0$$ ... ...

... BUT ... from where do we take the values of $$X$$ and $$Y$$ ... do they likewise come from $$K_0$$ ... or do they come from $$K$$ ...

Hope someone can help clarify this issue ...

Peter

 

[Euge]

Note $X$ and $Y$ are indeterminate variables. Elements of $K_0[X,Y]$ are polynomials in $X$ and $Y$ with coefficients in $K_0$. Since $K_0$ is a subset of $K$, the coefficients must belong to $K$.

 

[Math Amateur]

Note $X$ and $Y$ are indeterminate variables. Elements of $K_0[X,Y]$ are polynomials in $X$ and $Y$ with coefficients in $K_0$. Since $K_0$ is a subset of $K$, the coefficients must belong to $K$.

Thanks for your help Euge ...

Yes, I understand that $X$ and $Y$ are indeterminate variables, ... ... but then Kunz, in the examples following Definition 1.1, plots the zero sets i.e. 'plots' and displays $$f = 0$$ for a few curves, apparently taking values of $$X,Y$$ in $$K_0$$ ... but maybe you would say evaluating the points in $$K_0$$ ... ...

For example, see Example 1.2 (d) below where Kunz appears to take values of $$X, Y$$ in $$K_0 = \mathbb{R}$$ ... (indeed, slightly confusingly in a subset $$\mathbb{R} \subset \mathbb{R}_+$$ ... ... as follows:
View attachment 4557
Can you explain how one should view/understand these "plots" where $$X,Y$$ take values in $$K_0$$ ...

Peter

 

[Euge]

But wait, these graphs you've shown are traces of $\Bbb R$-rational points -- you take the zero set of the polynomial and intersect it with $\Bbb R^2$. So we substitute real values of $X$ and $Y$ to obtain the plots.

 

[Math Amateur]

But wait, these graphs you've shown are traces of $\Bbb R$-rational points -- you take the zero set of the polynomial and intersect it with $\Bbb R^2$. So we substitute real values of $X$ and $Y$ to obtain the plots.

Thanks Euge ... most helpful ...

And presumably, we could graph the trace of the zero set anyway ... essentially graphing the $$\mathbb{C}$$-rational points ... that is take $$K_0 = K = \mathbb{C}$$ ... (if we had 4 dimensions to hand ... :) ...)

Is that right?

Peter

 

[Euge]

You've got the idea. ;)

 

[Math Amateur]

Thanks Euge ... most helpful ...

And presumably, we could graph the trace of the zero set anyway ... essentially graphing the $$\mathbb{C}$$-rational points ... that is take $$K_0 = K = \mathbb{C}$$ ... (if we had 4 dimensions to hand ... :) ...)

Is that right?

Peter

Thanks for for all your help in this matter, Euge ...

Peter