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 (DZ-pairs). The paper is devoted to the study of DZ-pairs with rational coefficients. In our earlier paper [F. Pakovich and A. K. Zvonkin, Minimum degree of the difference of two polynomials over [Formula: see text], and weighted plane trees, Selecta Math., (N.S.) 20(4) (2014) 1003–1065], 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 [Formula: see text]. In our cited paper above, we classified all unitrees. In this paper, we compute all the corresponding polynomials. We also present some additional material concerning the Galois theory of DZ-pairs and weighted trees.
Davenport–Zannier polynomials over ℚ
