Diagonalization and rationalization of algebraic Laurent series

Adamczewski, Boris; Bell, Jason P.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 6, pp. 963-1004

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].

Nous démontrons une version quantitative d’un résultat de Furstenberg [20] et Deligne [14] : la diagonale d’une série formelle algébrique de plusieurs variables à coefficients dans un corps de caractéristique non nulle est une série formelle algébrique d’une variable. Comme conséquence, nous obtenons que, pour tout nombre premier $p$, la réduction modulo $p$ de la diagonale d’une série formelle algébrique de plusieurs variables $f$ à coefficients entiers est une série formelle algébrique de degré au plus $p^A$ et de hauteur au plus $A{p}^A$, où $A$ est une constante effective ne dépendant que du nombre de variables, du degré de $f$ et de la hauteur de $f$. Cela répond à une question soulevée par Deligne [14].

Classification : 13F25, 11B85, 11J85, 11T99, 34M99, 05A15, 33E99
Keywords: diagonals of algebraic functions, formal power series, multivariate Laurent series, G-functions, reduction modulo $p$
Mots-clés : diagonales de fonctions algébriques, séries formelles, séries de Laurent à plusieurs variables, G-fonctions, réduction modulo $p$

[1] B. Adamczewski & J. P. Bell, On vanishing coefficients of algebraic power series over fields of positive characteristic, Invent. Math. 187 (2012), 343-393. | MR

[2] J.-P. Allouche, Transcendence of formal power series with rational coefficients, Theoret. Comput. Sci. 218 (1999), 143-160. | MR

[3] J.-P. Allouche, D. Gouyou-Beauchamps & G. Skordev, Transcendence of binomial and Lucas' formal power series, J. Algebra 210 (1998), 577-592. | MR

[4] Y. André, $G$-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, 1989. | MR

[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] F. Beukers & C. A. M. Peters, A family of $K3$ surfaces and $\zeta \left(3\right)$, J. reine angew. Math. 351 (1984), 42-54. | MR

[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. | MR

[8] G. Christol, Diagonales de fractions rationnelles et équations différentielles, Groupe de travail d'analyse ultramétrique 10 (1982-1983), 1-10.

[9] G. Christol, Diagonales de fractions rationnelles et équations de Picard-Fuchs, Groupe de travail d'analyse ultramétrique 12 (1984-1985), 1-12. | MR | Numdam

[10] G. Christol, Diagonales de fractions rationnelles, in Séminaire de Théorie des Nombres, Paris 1986-87, Progr. Math. 75, Birkhäuser, 1988, 65-90. | MR

[11] G. Christol, Globally bounded solutions of differential equations, in Analytic number theory (Tokyo, 1988), Lecture Notes in Math. 1434, Springer, 1990, 45-64. | MR

[12] G. Christol, T. Kamae, M. Mendès France & G. Rauzy, Suites algébriques, automates et substitutions, Bull. Soc. Math. France 108 (1980), 401-419. | MR

[13] E. Delaygue, Arithmetic properties of Apéry-like numbers, preprint arXiv:1310.4131.

[14] P. Deligne, Intégration sur un cycle évanescent, Invent. Math. 76 (1984), 129-143.

[15] J. Denef & L. Lipshitz, Algebraic power series and diagonals, J. Number Theory 26 (1987), 46-67. | MR

[16] B. Dwork, G. Gerotto & F. J. Sullivan, An introduction to $G$-functions, Annals of Math. Studies 133, Princeton Univ. Press, 1994. | MR

[17] D. Eisenbud, Commutative algebra, Graduate Texts in Math. 150, Springer, 1995. | MR

[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. | MR | Numdam

[19] P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci. 49 (1987), 283-309. | MR

[20] H. Furstenberg, Algebraic functions over finite fields, J. Algebra 7 (1967), 271-277. | MR

[21] T. Harase, Algebraic elements in formal power series rings, Israel J. Math. 63 (1988), 281-288. | MR

[22] M. Kontsevich & D. Zagier, Periods, in Mathematics unlimited-2001 and beyond, Springer, 2001, 771-808. | MR

[23] L. Lipshitz, The diagonal of a $D$-finite power series is $D$-finite, J. Algebra 113 (1988), 373-378. | MR

[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. | MR

[25] P. Roquette, Einheiten und Divisorklassen in endlich erzeugbaren Körpern, Jber. Deutsch. Math. Verein 60 (1957), 1-21. | MR

[26] O. Salon, Suites automatiques à multi-indices, Séminaire de Théorie des Nombres de Bordeaux (1986-1987), exposé 4, 1-27. | MR | Numdam

[27] A. Sathaye, Generalized Newton-Puiseux expansion and Abhyankar-Moh semigroup theorem, Invent. Math. 74 (1983), 149-157. | MR

[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. | MR

[29] R. P. Stanley, Generating functions, in Studies in combinatorics, MAA Stud. Math. 17, Math. Assoc. America, 1978, 100-141. | MR

[30] R. P. Stanley, Differentiably finite power series, European J. Combin. 1 (1980), 175-188. | MR

[31] R. P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Math. 62, Cambridge Univ. Press, 1999. | MR

[32] M. Waldschmidt, Transcendence of periods: the state of the art, Pure Appl. Math. Q. 2 (2006), 435-463. | MR

[33] M. Waldschmidt, Elliptic functions and transcendence, in Surveys in number theory, Dev. Math. 17, Springer, 2008, 143-188. | MR

[34] C. F. Woodcock & H. Sharif, On the transcendence of certain series, J. Algebra 121 (1989), 364-369. | MR