Davenport Zannier polynomials over Q

摘要

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 Q, 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 Q. 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.