when is rational function regular?

Question

In general, how does one determine if a rational function is regular? I have the particular problem of determining in which points of the circle $V(x^2+y^2-1) \subseteq A^2$is the rational function $\alpha= \frac{y-1}{x}$ regular?

Answer 1

To say that $\phi= \frac{y-1}{x}$ is regular on the circle means that there exist polynomials $p(X,Y), q(X,Y)\in k[X,Y]$ such that: $$Y-1=X\cdot p(X,Y)+q(X,Y)\cdot(X^2+Y^2-1)\in k[X,Y]$$ But substituting $X=0$ in that equality yields $$Y-1=q(0,Y)\cdot(Y^2-1) \in k[Y] $$ which is impossible since the left hand side has degree $1$ whereas the right hand side is zero or has degree $\geq2$.
Hence the rational function $\phi$ is not regular.

Edit
Since $\phi$ is not regular but is clearly regular at all points of the circle different from $P=(0,-1)$, it follows that $P$ is the only point where $\phi$ is not regular, i.e. the only pole of $\phi$.