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


My question is as follows:

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

\(\displaystyle f = aX + bY + c \)

then \(\displaystyle a,b,c\) come from \(\displaystyle K_0\) ... that is \(\displaystyle a,b,c \in K_0\) ... ...

... BUT ... from where do we take the values of \(\displaystyle X\) and \(\displaystyle Y\) ... do they likewise come from \(\displaystyle K_0\) ... or do they come from \(\displaystyle K\) ...

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 \(\displaystyle f = 0\) for a few curves, apparently taking values of \(\displaystyle X,Y\) in \(\displaystyle K_0\) ... but maybe you would say evaluating the points in \(\displaystyle K_0\) ... ...

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

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 \(\displaystyle \mathbb{C}\)-rational points ... that is take \(\displaystyle K_0 = K = \mathbb{C}\) ... (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 \(\displaystyle \mathbb{C}\)-rational points ... that is take \(\displaystyle K_0 = K = \mathbb{C}\) ... (if we had 4 dimensions to hand ... :) ...)

Is that right?

Peter

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

Peter