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Razika Niboucha; Alain Salinier
Composition d’applications quasi-polynomiales
(Composition of quasi-polynomial maps)
Journal de théorie des nombres de Bordeaux, 29 no. 2 (2017), p. 569-601, doi: 10.5802/jtnb.992
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Class. Math.: 11B37
Keywords: Bijections entre entiers rationnels, coloriages, quasi-polynômes, suites récurrentes linéaires

Résumé - Abstract

The aim of this work is to study compositional properties of quasi-polynomial maps from $\mathbb{Z}$ to $\mathbb{Z}$, and more particularly of quasi-affine maps, namely quasi-polynomial maps of degree at most 1. We show that quasi-affine maps correspond to continuous endomorphisms of the algebra of recognizable bi-infinite sequences. We represent quasi-affine maps by means of finite sequences of integers, and we give explicit formulae on these sequences which translate composition or reversion of quasi-affine maps. Finally, we consider a coloring problem equivalent to the characterization of such sequences for quasi-affine bijections.

Bibliography

[1] Ahmed Aït-Mokhtar, Endomorphismes d’algèbres de suites, Ph. D. Thesis, Université de Limoges, France, 2008
[2] Ahmed Aït-Mokhtar, Applications purement semi-affines et tressages, C. R., Math., Acad. Sci. Paris 348 (2010), p. 1-4 Article |  MR 2586732
[3] Ahmed Aït-Mokhtar, Abdelkader Necer & Alain Salinier, Endomorphismes d’algèbres de suites, J. Théor. Nombres Bordx. 20 (2008), p. 1-21 Article
[4] Benali Benzaghou, Algèbres de Hadamard, Bull. Soc. Math. Fr. 98 (1970), p. 209-252 Article
[5] Jean Berstel & Christophe Reutenauer, Noncommutative rational series with applications, Encyclopedia of Mathematics and Its Applications 137, Cambridge University Press, 2011
[6] Nicolas Bourbaki, Eléments de mathématique. Algèbre Commutative. Chapitres 5 à 7, Springer, 2006
[7] Arthur Cayley, Researches on the Partition of Numbers, Philos. Trans. Roy. Soc. London 146 (1856), p. 127-140 Article
[8] Jean-Luc Chabert, Anneaux de Fatou, Enseign. Math. 18 (1972), p. 141-144
[9] Louis Comtet, Analyse combinatoire. Tome 1, Le mathématicien 4, Presses Universitaires de France, 1970
[10] Eugene Ehrhart, Polynômes arithmétiques et Méthode des Polyèdres en Combinatoire, International Series of Numerical Mathematics 35, Birkhäuser, 1977
[11] Ryszard Engelking, General topology. A revised and enlarged translation, Monografie Matematyczne. 60, PWN-Polish Scientific Publishers, 1977
[12] Graham Everest, Alf van der Poorten, Igor Shparlinski & Thomas Ward, Recurrence Sequences, Mathematical Surveys and Monographs 104, American Mathematical Society, 2003
[13] Georges Hansel, Une démonstration simple du théorème de Skolem-Mahler-Lech, Theor. Comput. Sci. 43 (1986), p. 91-98 Article
[14] Richard G. Larson & Earl J. Taft, The algebraic structure of linearly recursive sequences under Hadamard product, Isr. J. Math. 72 (1990), p. 118-132 Article
[15] Ivan Niven, The asymptotic density of sequences, Bull. Am. Math. Soc. 57 (1951), p. 420-434 Article
[16] George Pólya, Über ganzwertige ganze Funktionen, Palermo Rend. 40 (1915), p. 1-16 Article
[17] Laurent Schwartz, Analyse. 2e partie : Topologie générale et analyse fonctionnelle, Collection Enseignement des Sciences 11, Hermann, 1970
[18] Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, 1997, Corrected reprint of the 1986 hardback edition