Minimum Degree of the Difference of Two Polynomials over $\mathbb Q$. Part II: DavenportZannier pairs
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 DavenportZannier pairs, or DZpairs for short. The paper is devoted to the study of DZpairs with rational coefficients. In our earlier paper, in the framework of the theory of dessins d'enfants, we established a correspondence between DZpairs 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 DZpair 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 DZpairs and weighted trees.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 DOI:
 10.48550/arXiv.1509.07973
 arXiv:
 arXiv:1509.07973
 Bibcode:
 2015arXiv150907973P
 Keywords:

 Mathematics  Number Theory
 EPrint:
 This is the same version. The replacement is due to a bad appearance of some formulas