Skip to main content
SpringerLink
Log in

On vanishing coefficients of algebraic power series over fields of positive characteristic

  • Published:
Inventiones mathematicae Aims and scope

Abstract

Let K be a field of characteristic p>0 and let f(t 1,…,t d ) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K(t 1,…,t d ). We prove a generalization of both Derksen’s recent analogue of the Skolem–Mahler–Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices (n 1,…,n d )∈ℕd for which the coefficient of \(t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\) in f(t 1,…,t d ) is zero is a p-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to S-unit equations and more generally to the Mordell–Lang Theorem over fields of positive characteristic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allouche, J.-P., Cateland, E., Gilbert, W.J., Peitgen, H.-O., Shallit, J.O., Skordev, G.: Automatic maps in exotic numeration systems. Theory Comput. Syst. 30, 285–331 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Allouche, J.-P., Shallit, J.: Automatic Sequences. Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  3. Bézivin, J.-P.: Une généralisation du théorème de Skolem–Mahler–Lech. Q. J. Math. Oxf. 40, 133–138 (1989)

    Article  MATH  Google Scholar 

  4. Cerlienco, L., Mignotte, M., Piras, F.: Suites récurrentes linéaires : propriétés algébriques et arithmétiques. Enseign. Math. 33, 67–108 (1987)

    MathSciNet  MATH  Google Scholar 

  5. Christol, G.: Ensembles presque périodiques k-reconnaissables. Theor. Comput. Sci. 9, 141–145 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Christol, G., Kamae, T., Mendès France, M., Rauzy, G.: Suites algébriques, automates et substitutions. Bull. Soc. Math. Fr. 108, 401–419 (1980)

    MATH  Google Scholar 

  7. Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Math. Syst. Theory 3, 186–192 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deligne, P.: Intégration sur un cycle évanescent. Invent. Math. 76, 129–143 (1983)

    Article  MathSciNet  Google Scholar 

  9. Denef, J., Lipshitz, L.: Algebraic power series and diagonals. J. Number Theory 26, 46–67 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Derksen, H.: A Skolem-Mahler-Lech theorem in positive characteristic and finite automata. Invent. Math. 168, 175–224 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Derksen, H., Masser, D.: Linear equations over multiplicative groups, recurrences, and mixing I. Manuscript (2010)

  12. Eilenberg, S.: Automata, Languages, and Machines, vol. A. Academic Press, San Diego (1974)

    MATH  Google Scholar 

  13. Everest, G., van der Poorten, A., Shparlinski, I., Ward, T.: Recurrence Sequences. Mathematical Surveys and Monographs, vol. 104. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  14. Evertse, J.-H.: On sums of S-units and linear recurrences. Compos. Math. 53, 225–244 (1984)

    MathSciNet  MATH  Google Scholar 

  15. Evertse, J.-H., Györy, K., Stewart, C.L., Tijdeman, R.: S-unit equations and their applications. In: New Advances in Transcendence Theory, Durham, 1986, pp. 110–174. Cambridge Univ. Press, Cambridge (1988)

    Chapter  Google Scholar 

  16. Evertse, J.-H., Schlickewei, H.P., Schmidt, W.M.: Linear equations in variables which lie in a multiplicative group. Ann. Math. 155, 807–836 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Faltings, G.: Diophantine approximation on abelian varieties. Ann. Math. 133, 549–576 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Furstenberg, H.: Algebraic functions over finite fields. J. Algebra 7, 271–277 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ghioca, D.: The isotrivial case in the Mordell–Lang Theorem. Trans. Am. Math. Soc. 360, 3839–3856 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hansel, G.: Une démonstration simple du théorème de Skolem–Mahler–Lech. Theor. Comput. Sci. 43, 91–98 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Harase, T.: Algebraic elements in formal power series rings. Isr. J. Math. 63, 281–288 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Honkala, J.: A decision method for the recognizability of sets defined by number systems. Theor. Inform. Appl. 20, 395–403 (1986)

    MathSciNet  MATH  Google Scholar 

  23. Hrushovski, E.: The Mordell-Lang conjecture for function fields. J. Am. Math. Soc. 9, 667–690 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kedlaya, K.: Finite automata and algebraic extensions of function fields. J. Théor. Nombres Bordeaux 18, 379–420 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lang, S.: Integral points on curves. Publ. Math. IHÉS 6, 27–43 (1960)

    Google Scholar 

  26. Lech, C.: A note on recurring series. Ark. Mat. 2, 417–421 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mahler, K.: Zur Approximation algebraischer Zahlen, I. (Über den grössten Primteiler binärer Formen). Math. Ann. 107, 691–730 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mahler, K.: Eine arithmetische Eigenschaft der Taylor–Koeffizienten rationaler Funktionen. Proc. K. Ned. Akad. Wet. 38, 50–60 (1935)

    Google Scholar 

  29. Mahler, K.: On the Taylor coefficients of rational functions. Proc. Camb. Philos. Soc. 52, 39–48 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mahler, K.: Addendum to the paper “On the Taylor coefficients of rational functions”. Proc. Camb. Philos. Soc. 53, 544 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  31. Masser, D.: Mixing and linear equations over groups in positive characteristic. Isr. J. Math. 142, 189–204 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Moosa, R., Scanlon, T.: F-structures and integral points on semiabelian varieties over finite fields. Am. J. Math. 126, 473–522 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. van der Poorten, A.J.: Some facts that should be better known, especially about rational functions. In: Number Theory and Applications, Banff, AB, 1988. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, pp. 497–528. Kluwer Academic, Dordrecht (1989)

    Google Scholar 

  34. van der Poorten, A.J., Schlickewei, H.P.: Additive relations in fields. J. Aust. Math. Soc. 51, 154–170 (1991)

    Article  MATH  Google Scholar 

  35. Salon, O.: Suites automatiques à multi-indices et algébricité. C. R. Acad. Sci. Paris Sér. I Math. 305, 501–504 (1987)

    MathSciNet  MATH  Google Scholar 

  36. Schmidt, K.: The dynamics of algebraic Z d-actions. In: European Congress of Mathematics, vol. I, Barcelona, 2000. Progress in Mathematics, vol. 201, pp. 543–553. Birkhäuser, Basel (2001)

    Chapter  Google Scholar 

  37. Schmidt, K., Ward, T.: Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent. Math. 111, 69–76 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sharif, H., Woodcock, C.F.: Algebraic functions over a field of positive characteristic and Hadamard products. J. Lond. Math. Soc. 37, 395–403 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  39. Skolem, T.: Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und dio-phantischer Gleichungen. In: Comptes Rendus Congr. Math. Scand., Stockholm, pp. 163–188 (1934)

    Google Scholar 

  40. Tao, T.: Structure and Randomness, Pages from Year One of a Mathematical Blog. American Mathematical Society, Providence (2008)

    MATH  Google Scholar 

  41. Voloch, J.F.: The equation ax+by=1 in characteristic p. J. Number Theory 73, 195–200 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Camille Jordan, CNRS, Université de Lyon, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Boris Adamczewski

  2. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Jason P. Bell

Authors
  1. Boris Adamczewski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jason P. Bell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris Adamczewski.

Additional information

The First author was supported by ANR grants Hamot and SubTile. The second author was supported by NSERC grant 31-611456.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Adamczewski, B., Bell, J.P. On vanishing coefficients of algebraic power series over fields of positive characteristic. Invent. math. 187, 343–393 (2012). https://doi.org/10.1007/s00222-011-0337-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-011-0337-4

Keywords

Search

Navigation