7
$\begingroup$

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 $a'/(b'-z)$.

$\endgroup$
5
  • $\begingroup$ As long as you avoid letting $a\to0$, you can just take the distance from $a/(b-z)$ to $a'/(b'-z)$ to be $|b/a-b'/a'|$, I think. But if you want a nice metric topology that's okay for all $(a,b)\in\mathbb C^2$, there may well be a serious problem in the neighborhood the line $a=0$. $\endgroup$ Jul 19, 2015 at 12:38
  • $\begingroup$ I want a topology on the space of all rational functions of arbitrary denominator degree (and, if needed, a finite limit for $z\to\infty$) with properties that imply the above special cases. $\endgroup$ Jul 19, 2015 at 12:55
  • $\begingroup$ Not sure if it's quite what you need, but there's a paper by Laura DeMarco, "Iteration at the boundary of the space of rational maps", Duke Math. Journal. 130 (2005) 169-197, that might be relevant. In any case, I seem to recall that she considers some sort of inductive (projective?) limit of the space of rational functions of degree $d$ over all $d$. The paper is available here: math.northwestern.edu/~demarco/Duke_boundary.pdf $\endgroup$ Jul 19, 2015 at 13:37
  • $\begingroup$ @JoeSilverman: As far as I could see, the paper works throughout with constant degree. $\endgroup$ Jul 19, 2015 at 13:45
  • $\begingroup$ Okay, but since it's dealing with iteration, and the degree of $f^n$ is $(\deg f)^n$, somehow it needs to be relating maps of differing degrees. In any case, Laura would be a good person to ask about this. $\endgroup$ Jul 19, 2015 at 16:54

3 Answers 3

7
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ Thanks! The description on p. 292 in terms of seminorms is very useful! $\endgroup$ Jul 20, 2015 at 11:01
5
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ What would be the distance of two functions of the form $r(z):=a/(b−z)$? $\endgroup$ Jul 19, 2015 at 12:58
  • $\begingroup$ @GeraldEdgar It wasn't Arnold that made that remark. (Not that it matters.) $\endgroup$ Jul 19, 2015 at 13:15
  • 2
    $\begingroup$ The difficulty is that in this topology $z/(z-1/n)$ has no limit when $n\to\infty$. $\endgroup$ Jul 19, 2015 at 20:52
  • $\begingroup$ It seems in this topology, you cannot change the degree when you converge. Constants can converge to constants. Degree one maps (like $z/(z-1/n)$) cannot converge to a constant (like $1$). $\endgroup$ Jul 20, 2015 at 1:13
2
$\begingroup$

$\hbox{Rat}_d$, 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.

$\endgroup$
2
  • $\begingroup$ I need a single space independent of $d$. $\endgroup$ Jul 19, 2015 at 12:57
  • $\begingroup$ In that case, you can just view the union of all the spaces as a subset of $\mathbb C^\infty$. Thus a sequence of rational functions converges when its coefficients converge. $\endgroup$ Jul 20, 2015 at 2:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.