Natural topologies for the space of rational functions

Question

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and, if needed, a finite, or zero, limit for $z\to\infty$). Or spaces of meromorphic functions containing these functions.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a\ne 0$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.

If possible, I'd like to have a metric which gives a nice formula for the distance of $a/(b-z)$ to $a'/(b'-z)$.

Answer 1

Karl-Goswin Grosse-Erdmann, The locally convex topology on the space of meromorphic functions, J. Austr. Math. Soc., 59 (1995) 287-303

Answer 2

What happens with this way of doing it?

A rational function is a map from the Riemann sphere $\mathbb C \cup \{\infty\}$ to itself. The Riemann sphere is compact, so has a unique uniform structure. (So choose a nice metric for it, say the one from an actual sphere.) Use "uniform convergence".

This topology is metric.

Answer 3

$\hbox{Rat}_d$, the space of rational functions of degree $d$, is a $2d+1$ dimensional affine algebraic variety, so it embeds into $\mathbb A^N$ for some suffficiently large $N$. Since you're presumably interested in working over $\mathbb C$ (or maybe $\mathbb R)$, you can embed $\hbox{Rat}_d(\mathbb C)$ into $\mathbb C^N$ and then just use the usual topology on $\mathbb C^N$.

To be more precise, if you view a degree $d$ rational function $f(z)=F(z)/G(z)$ as the $2d+2$-tuple of the coefficients of $F$ and $G$, it is defined as a point in $\mathbb P^{2d+2}$, and $\hbox{Rat}_d$ is the complement of the resultant locus $\mathcal R:\hbox{Res}(F,G)=0$. The complement of a hypersurface such as $\mathcal R$ is an affine variety. There's a brief description of how this embedding works in (see especially Proposition 4.27)

The Arithmetic of Dynamical Systems, Springer, Section 4.3 "The Space $\hbox{Rat}_d$ of Rational Functions"

Addendum in answer to comment that "I need a single space independent of $d$"

Ah, that's much more problematic. Now you're into the whole yoga of degenerations of rational maps of degree $d$ on the boundary of their natural moduli space. For those who work in dynamical systems, it is more natural to look at the quotient space $\mathcal M_d := \hbox{Rat}_d/\hbox{PGL}_2$, where the action is via conjugation, $f^\phi=\phi^{-1}\circ f\circ\phi$. These spaces were studied by Milnor (Experimental Math 2(1), 37-83, 1993), who proves that $\mathcal M_d(\mathbb C)$ is an orbifold and that it has a reasonable compactification. In particular, for $d=2$ he proves that $\mathcal M_2(\mathbb C)\cong\mathbb C^2$, and its natural compactification is $\overline{\mathcal M_2}(\mathbb C)\cong\mathbb P^2(\mathbb C)$. So then you can take any natural metric on projective space. I took this up from the perspective of algebraic geometry (geometric invariant theory) in Duke Math J. 94(1), 41-77, 1998. Of course, this may not be what you need, but it provides one approach. The point is that if you want a nice compactification of a moduli space, you usually need to restrict what sort of degenerations are allowed. In GIT language, the magic word is semi-stability.

Answer 4

The most important work on this problem seems to belong to J.H. Williamson, On topologising the field C(t), Proc. Amer. Math. Soc. 5 (1954), 729–734. The message of the paper, however, is not very encouraging: there are some possible topologies, but they cannot have too many desirable properties at once.