Minimum degree of the difference of two polynomials over ${\mathbb Q}$, and weighted plane trees

Abstract

A weighted bicolored plane tree (or just tree for short) is a bicolored plane tree whose edges are endowed with positive integral weights. The degree of a vertex is defined as the sum of the weights of the edges incident to this vertex. Using the theory of dessins d’enfants, which studies the action of the absolute Galois group on graphs embedded into Riemann surfaces, we show that a weighted plane tree is a graphical representation of a pair of coprime polynomials $P,Q\in {\mathbb C}\,[x]$ such that: (a) $\deg P = \deg Q$, and $P$ and $Q$ have the same leading coefficient; (b) the multiplicities of the roots of $P$ (respectively, of $Q$) are equal to the degrees of the black (respectively, white) vertices of the corresponding tree; (c) the degree of the difference $P-Q$ attains the minimum which is possible for the given multiplicities of the roots of $P$ and $Q$. Moreover, if a tree in question is uniquely determined by the set of its black and white vertex degrees (we call such trees unitrees), then the corresponding polynomials are defined over ${\mathbb Q}$. The pairs of polynomials $P,Q$ such that the degree of the difference $P-Q$ attains the minimum, and especially those defined over ${\mathbb Q}$, are related to some important questions of number theory. Dozens of papers, from 1965 (Birch et al. in Norske Vid Selsk Forh 38:65–69, 1965) to 2010 (Beukers and Stewart in J Number Theory 130:660–679, 2010), were dedicated to their study. The main result of this paper is a complete classification of the unitrees, which provides us with the most massive class of such pairs defined over ${\mathbb Q}$. We also study combinatorial invariants of the Galois action on trees, as well as on the corresponding polynomial pairs, which permit us to find yet more examples defined over ${\mathbb Q}$. In a subsequent paper, we compute the polynomials $P,Q$ corresponding to all the unitrees.