J. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Science, vol.98, issue.2, pp.163-197, 1992.
DOI : 10.1016/0304-3975(92)90001-V

A. Bostan, G. Christol, and P. Dumas, Fast computation of the N th term of an algebraic series over a finite prime field, ISSAC'16, pp.119-126, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01389421

A. Bostan, P. Gaudry, and . Schost, Linear Recurrences with Polynomial Coefficients and Application to Integer Factorization and Cartier???Manin Operator, SIAM Journal on Computing, vol.36, issue.6, pp.1777-1806, 2007.
DOI : 10.1137/S0097539704443793
URL : https://hal.archives-ouvertes.fr/inria-00103401

A. Bostan, C. Jeannerod, and E. Schost, Solving structured linear systems with large displacement rank, Theoretical Computer Science, vol.407, issue.1-3, pp.155-181, 2008.
DOI : 10.1016/j.tcs.2008.05.014
URL : https://doi.org/10.1016/j.tcs.2008.05.014

A. Bridy, Automatic sequences and curves over finite fields, Algebra & Number Theory, vol.254, issue.3, pp.685-712, 2017.
DOI : 10.1007/978-0-387-09494-6
URL : http://arxiv.org/pdf/1604.08241

G. , Ensembles presque periodiques k-reconnaissables, Theoretical Computer Science, vol.9, issue.1, pp.141-145, 1979.
DOI : 10.1016/0304-3975(79)90011-2

G. Christol, T. Kamae, M. M. France, and G. Rauzy, Suites alg??briques, automates et substitutions, Bulletin de la Société mathématique de France, vol.79, issue.4, pp.401-419, 1980.
DOI : 10.24033/bsmf.1926
URL : http://www.numdam.org/article/BSMF_1980__108__401_0.pdf

D. V. Chudnovsky and G. V. Chudnovsky, On expansion of algebraic functions in power and Puiseux series, I, Journal of Complexity, vol.2, issue.4, pp.271-294, 1986.
DOI : 10.1016/0885-064X(86)90006-3

D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan, Ramanujan revisited, pp.375-472, 1987.
DOI : 10.1007/978-1-4757-2736-4_63

D. V. Chudnovsky and G. V. Chudnovsky, Computer algebra in the service of mathematical physics and number theory, Computers in mathematics (Stanford, pp.109-232, 1986.

P. Deligne, Int???gration sur un cycle ???vanescent, Inventiones Mathematicae, vol.38, issue.1, pp.129-143, 1984.
DOI : 10.1007/BF01388496

D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol.150, 1995.

C. M. Fiduccia, An Efficient Formula for Linear Recurrences, SIAM Journal on Computing, vol.14, issue.1, pp.106-112, 1985.
DOI : 10.1137/0214007

H. Furstenberg, Algebraic functions over finite fields, Journal of Algebra, vol.7, issue.2, pp.271-277, 1967.
DOI : 10.1016/0021-8693(67)90061-0
URL : https://doi.org/10.1016/0021-8693(67)90061-0

T. Harase, Algebraic elements in formal power series rings, Israel Journal of Mathematics, vol.46, issue.3, pp.281-288, 1988.
DOI : 10.1007/BF02778035

H. T. Kung and J. F. Traub, All Algebraic Functions Can Be Computed Fast, Journal of the ACM, vol.25, issue.2, pp.245-260, 1978.
DOI : 10.1145/322063.322068
URL : http://repository.cmu.edu/cgi/viewcontent.cgi?article=3139&context=compsci

V. Y. Pan, Structured matrices and polynomials, Birkhäuser Boston, Inc, 2001.
DOI : 10.1007/978-1-4612-0129-8
URL : https://hal.archives-ouvertes.fr/inria-00073171

D. Speyer, Christol's theorem and the Cartier operator Blog post, 11 Feb, Secret Blogging Seminar, vol.11, issue.02, 2010.

F. E. Inria, -mail address: alin.bostan@inria.fr CNRS, France E-mail address: xavier.caruso@normalesup.org IMJ, France E-mail address: christol.gilles@gmail.com Inria