Title: Davenport-Zannier polynomials and dessins d'enfants /
Authors: Adrianov, Nikolai M., 1973-; Pakovich, Fedor, 1970-; Zvonkin, A. K. 1948-
Year: 2020
Subjects: Dessins d'enfants (Mathematics); Arithmetical algebraic geometry; Trees (Graph theory); Galois theory; Polynomials; Algebraic fields; Algebraic fields; Arithmetical algebraic geometry; Dessins d'enfants (Mathematics); Galois theory; Polynomials; Trees (Graph theory); Number theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Dessins d'enfants, Belyĭ theory.; Combinatorics -- Graph theory -- Trees.; Combinatorics -- Graph theory -- Planar graphs; geometric and topological aspect.; Combinatorics -- Graph theory -- Signed and weighted graphs.; Combinatorics -- Algebraic combinatorics -- Group actions on combinatorial structures.; Number theory -- Algebraic number theory: global fields -- Galois theory.; Field theory and polynomials -- General field theory -- Special polynomials.; Field theory and polynomials -- Computational aspects of field theory and polynomials -- Computational aspects of field theory and polynomials.; Group theory and generalizations -- Permutation groups -- Primitive groups.
Abstract: The French expression "dessins d'enfants" means children's drawings. This term was coined by the great French mathematician Alexandre Grothendieck in order to denominate a method of pictorial representation of some highly interesting classes of polynomials and rational functions. The polynomials studied in this book take their origin in number theory. The authors show how, by drawing simple pictures, one can prove some long-standing conjectures and formulate new ones. The theory presented here touches upon many different fields of mathematics.
Contents: Introduction -- Dessins d'enfants : from polynomials through Belyĭ functions to weighted trees -- Existence theorem -- Recapitulation and perspectives -- Classification of unitrees -- Computation of Davenport-Zannier pairs for unitrees -- Primitive monodromy groups of weighted trees -- Trees with primitive monodromy groups -- A zoo of examples and constructions -- Diophantine invariants -- Enumeration -- What remains to be done.