Pairs of tree dessins, their Shabat polynomials, and monodromy groups

TL;DR

This work classifies all passports that determine exactly two tree dessins and computes their Shabat polynomials and monodromy groups. Using differentiation tricks, brush compositions, and controlled polynomial constructions, it treats six infinite families F1–F6 and twelve sporadic cases F7–F12, detailing fields of definition and the imprimitive nature of the monodromy groups. The results extend existing catalogs of Belyi maps and provide insight into the Galois action on dessins d'enfants by contrasting split versus fixed-point cases. The methods yield explicit algebraic certificates for the two-tree phenomenon and offer a foundation for exploring more complex passport configurations and invariants.

Abstract

Coverings of the Riemann sphere by itself, ramified over two points, are given by so-called Shabat polynomials. The correspondence between Grothendieck's dessins d'enfants and Belyi maps then implies a bijection between Shabat polynomials and tree dessins (bicolored plane trees). Dessins can be assigned a combinatorial invariant known as their passport, which records the degrees of their vertices. We consider all possible passports determining a pair of tree dessins, determining the associated Shabat polynomials and monodromy groups.

Figures (16)

• Figure 1: The families of passports with exactly two trees.
• Figure 2: Trees in $\mathcal{F}_{1}$ with $r=3,s=5$ and $t=6$.
• Figure 3: Trees in $\mathcal{F}_{2}$ with $r=3$ and $s=5$.
• Figure 4: The tree corresponding to $R$ in \ref{['lem:p2alt']}.
• Figure 5: Trees for $\mathcal{F}_{3}$ with $r=3$ and $s=5$.

Theorems & Definitions (42)

• Definition 1: Equivalence of Shabat polynomials
• Lemma 2.1: couveignes1994calcul
• Lemma 2.2
• Theorem 2.3: ritt1922prime
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1
• proof