[Paper Review] Minimum Degree of the Difference of Two Polynomials over $\mathbb Q$. Part II: Davenport-Zannier pairs

Fedor Pakovich, Alexander K. Zvonkin|arXiv (Cornell University)|Sep 26, 2015

Algebraic Geometry and Number Theory10 references6 citations

TL;DR

This paper completes the classification of Davenport–Zannier pairs with rational coefficients by computing all polynomials corresponding to unitrees—weighted bicolored plane trees uniquely determined by their vertex degree partitions. Using the dessins d'enfants correspondence, the authors derive explicit polynomial expressions for all 10 infinite series (A–J) and 10 sporadic cases (K–T), proving their rationality and analyzing Galois invariants, thus resolving the minimum degree problem for polynomial differences over ℚ.

ABSTRACT

In this paper we study pairs of polynomials with a given factorization pattern and such that the degree of their difference attains its minimum. We call such pairs of polynomials Davenport--Zannier pairs, or DZ-pairs for short. The paper is devoted to the study of DZ-pairs with rational coefficients. In our earlier paper, in the framework of the theory of dessins d'enfants, we established a correspondence between DZ-pairs and weighted bicolored plane trees. These are bicolored plane trees whose edges are endowed with positive integral weights. When such a tree is uniquely determined by the set of black and white degrees of its vertices, it is called unitree, and the corresponding DZ-pair is defined over $\mathbb Q$. In our earlier paper, we classified all unitrees. In this paper, we compute all the corresponding polynomials. In the final part of the paper we present some additional material concerning the Galois theory of DZ-pairs and weighted trees.

Motivation & Objective

• To compute all polynomials corresponding to unitrees, which are weighted bicolored plane trees uniquely determined by their vertex degree partitions.
• To establish that Davenport–Zannier pairs associated with unitrees are defined over ℚ, extending the classification from prior work.
• To analyze the Galois action on DZ-pairs and their corresponding weighted trees, identifying invariants and splitting behaviors.
• To provide explicit analytic expressions for all DZ-pairs in infinite series A–J and sporadic cases K–T.
• To explore cases where the minimum degree condition is relaxed, yielding rational solutions despite non-minimal difference degrees.

Proposed method

• Leverage the dessins d'enfants correspondence between Davenport–Zannier pairs and weighted bicolored plane trees with prescribed vertex degrees.
• Use the classification of unitrees from prior work [17] as the foundation for constructing all rational DZ-pairs.
• Derive explicit polynomial expressions for each DZ-pair using Belyi functions and symmetric rational maps.
• Verify rationality of the constructed polynomials by checking that their roots and coefficients lie in ℚ.
• Analyze Galois invariants by studying the monodromy action on the trees and the splitting behavior of combinatorial orbits.
• Construct examples with non-minimal difference degrees but rational coefficients, using tree modifications and cube constructions.

Experimental results

Research questions

• RQ1 Which Davenport–Zannier pairs with rational coefficients correspond to unitrees, and what are their explicit polynomial forms?
• RQ2 How does the Galois group act on DZ-pairs and their associated weighted trees, and what invariants emerge from this action?
• RQ3 Can rational DZ-pairs be constructed even when the difference degree exceeds the theoretical minimum, and under what conditions?
• RQ4 Why do certain weighted trees with identical passports yield DZ-pairs defined over ℚ despite non-trivial symmetry breaking?
• RQ5 What role do combinatorial orbits of trees play in determining the field of definition of DZ-pairs, especially in cases with split Galois actions?

Key findings

• All Davenport–Zannier pairs corresponding to the 10 infinite series (A–J) and 10 sporadic trees (K–T) are explicitly constructed and shown to be defined over ℚ.
• The DZ-pair for passport (9⁵, 5⁹) is defined over ℚ, despite all known Galois invariants failing to explain this phenomenon.
• An infinite family of trees with passport (k², 4¹1²ᵏ⁻⁴) yields two non-isomorphic trees—one symmetric, one asymmetric—each giving a DZ-pair defined over ℚ.
• For k=7, a DZ-pair with degree 9 difference (exceeding the minimum k+1=8) is constructed with both A and B defined over ℚ.
• For k=6, a DZ-pair with a multiple root in the difference polynomial (degree 9 > minimum 7) is constructed, with explicit rational coefficients.
• The Belyi functions for symmetric and asymmetric trees in the (k², 4¹1²ᵏ⁻⁴) family are explicitly given, and their correctness is verified via vanishing derivatives at x=0.