# Conservative polynomials and yet another action of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on plane trees

Fedor Pakovich^{[1]}

- [1] Department of Mathematics Ben Gurion University Beer Sheva 84105 P.O.B. 653, , Israel

Journal de Théorie des Nombres de Bordeaux (2008)

- Volume: 20, Issue: 1, page 205-218
- ISSN: 1246-7405

## Access Full Article

top## Abstract

top## How to cite

topPakovich, Fedor. "Conservative polynomials and yet another action of $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on plane trees." Journal de Théorie des Nombres de Bordeaux 20.1 (2008): 205-218. <http://eudml.org/doc/10829>.

@article{Pakovich2008,

abstract = {In this paper we study an action $D$ of the absolute Galois group $\Gamma =\operatorname\{Gal\}(\bar\{\mathbb\{Q\}\}/\mathbb\{Q\})$ on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action $D$ is induced by the action of $\Gamma $ on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action $D$ and compare it with the Grothendieck action.},

affiliation = {Department of Mathematics Ben Gurion University Beer Sheva 84105 P.O.B. 653, , Israel},

author = {Pakovich, Fedor},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {action of absolute Galois group on bicolored plane trees; Grothendieck action},

language = {eng},

number = {1},

pages = {205-218},

publisher = {Université Bordeaux 1},

title = {Conservative polynomials and yet another action of $\operatorname\{Gal\}(\bar\{\mathbb\{Q\}\}/\mathbb\{Q\})$ on plane trees},

url = {http://eudml.org/doc/10829},

volume = {20},

year = {2008},

}

TY - JOUR

AU - Pakovich, Fedor

TI - Conservative polynomials and yet another action of $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on plane trees

JO - Journal de Théorie des Nombres de Bordeaux

PY - 2008

PB - Université Bordeaux 1

VL - 20

IS - 1

SP - 205

EP - 218

AB - In this paper we study an action $D$ of the absolute Galois group $\Gamma =\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action $D$ is induced by the action of $\Gamma $ on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action $D$ and compare it with the Grothendieck action.

LA - eng

KW - action of absolute Galois group on bicolored plane trees; Grothendieck action

UR - http://eudml.org/doc/10829

ER -

## References

top- G. Belyi, On Galois extensions of a maximal cyclotomic field. Math. USSR, Izv. 14 (1980), 247–256. Zbl0429.12004MR534593
- A. Douady, J. Hubbard, A proof of Thurston’s topological characterization of rational functions. Acta Math. 171, No.2 (1993), 263-297. Zbl0806.30027
- A. Kostrikin, Conservative polynomials. In “Stud. Algebra Tbilisi”, 115–129, 1984. Zbl0728.12003
- S. Lando, A. Zvonkin, Graphs on Surfaces and Their Applications. Encyclopedia of Mathematical Sciences 141(II), Berlin: Springer, 2004. Zbl1040.05001MR2036721
- K. Pilgrim, Dessins d’enfants and Hubbard trees, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 5, 671–693. Zbl1066.14503
- A. Poirier, On postcritically finite polynomials, part 1: critical portraits. Preprint, arxiv:math. DS/9305207. Zbl1179.37066
- A. Poirier, On postcritically finite polynomials, part 2: Hubbard trees. Preprint, arxiv:math. DS/9307235.
- L. Schneps, Dessins d’enfants on the Riemann sphere. In “The Grothendieck Theory of Dessins D’enfants” (L. Shneps eds.), Cambridge University Press, London mathematical society lecture notes series 200 (1994), 47–77. Zbl0823.14017
- J. Silverman, The field of definition for dynamical systems on ${\mathbb{P}}^{1}$. Compos. Math. 98, No.3 (1995), 269–304. Zbl0849.11090MR1351830
- S. Smale, The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. 4 (1981), 1–36. Zbl0456.12012MR590817
- D. Tischler, Critical points and values of complex polynomials. J. of Complexity 5 (1989), 438–456. Zbl0728.12004MR1028906

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.