[Paper Review] Rationality in map and hypermap enumeration by genus

Maxim Kazarian, Peter Zograf|arXiv (Cornell University)|Sep 18, 2016

Advanced Combinatorial Mathematics12 references3 citations

TL;DR

This paper establishes that generating functions for rooted maps and hypermaps of fixed genus become rational after a simple change of variables: $ s = t(1-2t) $ for hypermaps and $ s = t(1-3t) $ for maps. The rational forms reveal polynomial numerators with integer coefficients obeying differential recursions, and denominators as products of powers of linear terms, providing a deep algebraic structure to enumeration problems in topological graph theory and Grothendieck's dessins d'enfants.

ABSTRACT

Generating functions for a fixed genus map and hypermap enumeration become rational after a simple explicit change of variables. Their numerators are polynomials with integer coefficients that obey a differential recursion, and denominators are products of powers of explicit linear functions.

Motivation & Objective

• To uncover the rational structure underlying genus-generating functions for rooted maps and hypermaps.
• To demonstrate that a simple change of variables transforms these generating functions into rational functions.
• To characterize the polynomial numerators and rational denominators that emerge after substitution.
• To establish recursive differential equations governing the coefficients of the rational forms.
• To connect the algebraic structure to known results in topological recursion and dessins d'enfants enumeration.

Proposed method

• Apply the substitution $ s = t(1-2t) $ for hypermaps and $ s = t(1-3t) $ for maps to transform the generating functions.
• Use the differential recursion derived from the KP equation for hypermap and map enumeration to derive ODEs in the new variable $ t $.
• Solve the resulting ordinary differential equations recursively, starting from the base case $ g=0 $, using integrating factors.
• Express the generating functions as rational functions by decomposing the integrand into partial fractions with poles at $ 1-2t $ and $ 1-4t $ (for hypermaps), or $ 1-2t $ and $ 1-6t $ (for maps).
• Prove that the resulting numerators are polynomials with integer coefficients by induction and coefficient recursion.
• Verify the rational form by showing the generating function satisfies the transformed ODE and initial conditions.

Experimental results

Research questions

• RQ1 Can the generating functions for rooted hypermaps of genus $ g $ be made rational via a simple change of variables?
• RQ2 What is the algebraic structure of the rationalized generating functions for maps and hypermaps by genus?
• RQ3 Do the numerators of the rationalized generating functions satisfy a differential recursion?
• RQ4 How do the denominators of the rationalized generating functions relate to the topological invariants of the maps?
• RQ5 Can the rational form be used to efficiently compute higher-genus enumeration coefficients?

Key findings

• After the substitution $ s = t(1-2t) $, the genus-0 hypermap generating function becomes $ C_0(t(1-2t)) = \frac{t(1-3t)}{(1-2t)^2} $, a rational function.
• For $ g \geq 2 $, $ C_g(t(1-2t)) = \frac{P_g(t)}{(1-t)^{4g-3}(1-4t)^{5g-3}} $, where $ P_g(t) $ is a polynomial with integer coefficients and leading term $ \frac{(2g)!}{g+1} t^{2g+1} $.
• The genus-1 hypermap generating function satisfies $ C_1(t(1-2t)) = \frac{t^3}{(1-t)(1-4t)^2} $, confirming the rational form.
• For maps, the substitution $ s = t(1-3t) $ rationalizes the generating function, yielding $ \widetilde{C}_0(t(1-3t)) = \frac{1-4t}{(1-3t)^2} $.
• The rational form for $ \widetilde{C}_g(t(1-3t)) $ is $ \frac{\widetilde{P}_g(t)}{(1-2t)^{3g-3}(1-6t)^{5g-3}} $, with $ \widetilde{P}_g(t) $ a polynomial with integer coefficients.
• The coefficients of the numerator polynomials satisfy a recursive differential relation derived from the KP hierarchy, enabling algorithmic computation.