Conservative polynomials and yet another action of Gal ( ℚ ¯ / 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

Abstract

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In this paper we study an action D of the absolute Galois group Γ = Gal ( ℚ ¯ / 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 Γ 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.

How to cite

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Pakovich, 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

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  1. G. Belyi, On Galois extensions of a maximal cyclotomic field. Math. USSR, Izv. 14 (1980), 247–256. Zbl0429.12004MR534593
  2. A. Douady, J. Hubbard, A proof of Thurston’s topological characterization of rational functions. Acta Math. 171, No.2 (1993), 263-297. Zbl0806.30027
  3. A. Kostrikin, Conservative polynomials. In “Stud. Algebra Tbilisi”, 115–129, 1984. Zbl0728.12003
  4. S. Lando, A. Zvonkin, Graphs on Surfaces and Their Applications. Encyclopedia of Mathematical Sciences 141(II), Berlin: Springer, 2004. Zbl1040.05001MR2036721
  5. K. Pilgrim, Dessins d’enfants and Hubbard trees, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 5, 671–693. Zbl1066.14503
  6. A. Poirier, On postcritically finite polynomials, part 1: critical portraits. Preprint, arxiv:math. DS/9305207. Zbl1179.37066
  7. A. Poirier, On postcritically finite polynomials, part 2: Hubbard trees. Preprint, arxiv:math. DS/9307235. 
  8. 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
  9. J. Silverman, The field of definition for dynamical systems on ℙ 1 . Compos. Math. 98, No.3 (1995), 269–304. Zbl0849.11090MR1351830
  10. S. Smale, The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. 4 (1981), 1–36. Zbl0456.12012MR590817
  11. D. Tischler, Critical points and values of complex polynomials. J. of Complexity 5 (1989), 438–456. Zbl0728.12004MR1028906

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