# Diagonalization and rationalization of algebraic Laurent series

• Volume: 46, Issue: 6, page 963-1004
• ISSN: 0012-9593

top

## Abstract

top
We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most ${p}^{A}$ and height at most $A{p}^{A}$, where $A$ is an effective constant that only depends on the number of variables, the degree of $f$ and the height of $f$. This answers a question raised by Deligne [14].

## How to cite

top

Adamczewski, Boris, and Bell, Jason P.. "Diagonalization and rationalization of algebraic Laurent series." Annales scientifiques de l'École Normale Supérieure 46.6 (2013): 963-1004. <http://eudml.org/doc/272194>.

abstract = {We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most $p^\{A\}$ and height at most $Ap^\{A\}$, where $A$ is an effective constant that only depends on the number of variables, the degree of $f$ and the height of $f$. This answers a question raised by Deligne [14].},
author = {Adamczewski, Boris, Bell, Jason P.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {diagonals of algebraic functions; formal power series; multivariate Laurent series; G-functions; reduction modulo $p$},
language = {eng},
number = {6},
pages = {963-1004},
publisher = {Société mathématique de France},
title = {Diagonalization and rationalization of algebraic Laurent series},
url = {http://eudml.org/doc/272194},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Bell, Jason P.
TI - Diagonalization and rationalization of algebraic Laurent series
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 6
SP - 963
EP - 1004
AB - We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most $p^{A}$ and height at most $Ap^{A}$, where $A$ is an effective constant that only depends on the number of variables, the degree of $f$ and the height of $f$. This answers a question raised by Deligne [14].
LA - eng
KW - diagonals of algebraic functions; formal power series; multivariate Laurent series; G-functions; reduction modulo $p$
UR - http://eudml.org/doc/272194
ER -

## References

top
1. [1] B. Adamczewski & J. P. Bell, On vanishing coefficients of algebraic power series over fields of positive characteristic, Invent. Math.187 (2012), 343–393. Zbl1257.11027MR2885622
2. [2] J.-P. Allouche, Transcendence of formal power series with rational coefficients, Theoret. Comput. Sci.218 (1999), 143–160. Zbl0916.68123MR1687784
3. [3] J.-P. Allouche, D. Gouyou-Beauchamps & G. Skordev, Transcendence of binomial and Lucas’ formal power series, J. Algebra210 (1998), 577–592. Zbl0980.11030MR1662292
4. [4] Y. André, $G$-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, 1989. MR990016
5. [5] F. Beukers, Congruence properties of coefficients of solutions of Picard-Fuchs equations, Groupe de travail d’analyse ultramétrique 14 (1986–1987), 1–6.
6. [6] F. Beukers & C. A. M. Peters, A family of $K3$ surfaces and $\zeta \left(3\right)$, J. reine angew. Math. 351 (1984), 42–54. Zbl0541.14007MR749676
7. [7] R. H. Cameron & W. T. Martin, Analytic continuation of diagonals and Hadamard compositions of multiple power series, Trans. Amer. Math. Soc.44 (1938), 1–7. Zbl0019.07102MR1501956
8. [8] G. Christol, Diagonales de fractions rationnelles et équations différentielles, Groupe de travail d’analyse ultramétrique 10 (1982–1983), 1–10.
9. [9] G. Christol, Diagonales de fractions rationnelles et équations de Picard-Fuchs, Groupe de travail d’analyse ultramétrique 12 (1984–1985), 1–12. MR848993
10. [10] G. Christol, Diagonales de fractions rationnelles, in Séminaire de Théorie des Nombres, Paris 1986–87, Progr. Math. 75, Birkhäuser, 1988, 65–90. MR990506
11. [11] G. Christol, Globally bounded solutions of differential equations, in Analytic number theory (Tokyo, 1988), Lecture Notes in Math. 1434, Springer, 1990, 45–64. MR1071744
12. [12] G. Christol, T. Kamae, M. Mendès France & G. Rauzy, Suites algébriques, automates et substitutions, Bull. Soc. Math. France108 (1980), 401–419. Zbl0472.10035MR614317
13. [13] E. Delaygue, Arithmetic properties of Apéry-like numbers, preprint arXiv:1310.4131. Zbl1297.11072
14. [14] P. Deligne, Intégration sur un cycle évanescent, Invent. Math.76 (1984), 129–143. Zbl0538.13007
15. [15] J. Denef & L. Lipshitz, Algebraic power series and diagonals, J. Number Theory26 (1987), 46–67. Zbl0609.12020MR883533
16. [16] B. Dwork, G. Gerotto & F. J. Sullivan, An introduction to $G$-functions, Annals of Math. Studies 133, Princeton Univ. Press, 1994. Zbl0830.12004MR1274045
17. [17] D. Eisenbud, Commutative algebra, Graduate Texts in Math. 150, Springer, 1995. MR1322960
18. [18] S. Fischler, Irrationalité de valeurs de zêta (d’après Apéry, Rivoal, ...), Séminaire Bourbaki, vol. 2002/03, exp. no 910, Astérisque 294 (2004), 27–62. Zbl1101.11024MR2111638
19. [19] P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci.49 (1987), 283–309. Zbl0612.68069MR909335
20. [20] H. Furstenberg, Algebraic functions over finite fields, J. Algebra7 (1967), 271–277. Zbl0175.03903MR215820
21. [21] T. Harase, Algebraic elements in formal power series rings, Israel J. Math.63 (1988), 281–288. Zbl0675.13015MR969943
22. [22] M. Kontsevich & D. Zagier, Periods, in Mathematics unlimited—2001 and beyond, Springer, 2001, 771–808. MR1852188
23. [23] L. Lipshitz, The diagonal of a $D$-finite power series is $D$-finite, J. Algebra113 (1988), 373–378. Zbl0657.13024MR929767
24. [24] L. Lipshitz & A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic, in Number theory (Banff, AB, 1988), de Gruyter, 1990, 339–358. MR1106672
25. [25] P. Roquette, Einheiten und Divisorklassen in endlich erzeugbaren Körpern, Jber. Deutsch. Math. Verein60 (1957), 1–21. Zbl0079.26901MR104652
26. [26] O. Salon, Suites automatiques à multi-indices, Séminaire de Théorie des Nombres de Bordeaux (1986–1987), exposé 4, 1–27. MR1050262
27. [27] A. Sathaye, Generalized Newton-Puiseux expansion and Abhyankar-Moh semigroup theorem, Invent. Math.74 (1983), 149–157. Zbl0549.14001MR722730
28. [28] H. Sharif & C. F. Woodcock, Algebraic functions over a field of positive characteristic and Hadamard products, J. London Math. Soc.37 (1988), 395–403. Zbl0612.12018MR939116
29. [29] R. P. Stanley, Generating functions, in Studies in combinatorics, MAA Stud. Math. 17, Math. Assoc. America, 1978, 100–141. MR513004
30. [30] R. P. Stanley, Differentiably finite power series, European J. Combin.1 (1980), 175–188. Zbl0445.05012MR587530
31. [31] R. P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Math. 62, Cambridge Univ. Press, 1999. Zbl0928.05001MR1676282
32. [32] M. Waldschmidt, Transcendence of periods: the state of the art, Pure Appl. Math. Q.2 (2006), 435–463. Zbl1220.11090MR2251476
33. [33] M. Waldschmidt, Elliptic functions and transcendence, in Surveys in number theory, Dev. Math. 17, Springer, 2008, 143–188. MR2462949
34. [34] C. F. Woodcock & H. Sharif, On the transcendence of certain series, J. Algebra121 (1989), 364–369. Zbl0689.13014MR992771

## NotesEmbed?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.