I'm learning algebraic geometry and recently found out that every function regular on all of an affine variety $X$ can be identified with a polynomial on $X$. I was wondering if it's always possible or feasible to match a regular function written as a rational function with a corresponding polynomial.
For example, if we let $X$ be $Z(x – 1)$ and consider the ring of polynomials $\mathbb{R} [x]$, then an example of a polynomial corresponding to $\frac{1}{x}$ is $1$ since they have the same value on $Z(x – 1)$ (both are $1$ when $x = 1$).
I have been unable to do the same with other functions, however. If we let $X$ be $Z(xy – 1)$ and consider the ring of polynomials $\mathbb{R} [x, y]$, I haven't been able to find a polynomial matching with $\frac{1}{x + y}$. I've tried multiplying the numerator and denominator by things such as $x$ and $x + y$, but I still haven't been able to get it into a polynomial form or find a matching polynomial. Is there a general way to do this, or at least one that can help in this situation?
Any hints or help would be appreciated!
Best Answer
You can't do this, because $\frac{1}{x+y}$ is not regular on $V(xy-1)$: it has a pole at $(\pm i,\mp i)$. One thing which you'll learn as you progress in algebraic geometry is that varieties can see points defined over field extensions - this is perhaps unintuitive at first, but you'll find that it's actually a wonderful thing as you learn a bit more. Right now, what this means for you is that you have to check regularity of your function at complex points too, and here you find an issue with the function $\frac{1}{x+y}$.