Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Polynomial values in affine subspaces of finite fields

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previously known results, and also extend the range of such results to polynomials of degrees higher than the characteristic of the field. Our approach is based on estimates for a certain new type of exponential sums. The results we obtain have a wide scope of applications similar to those associated with their counterparts studying consecutive intervals over prime fields instead of affine subspaces. Here we give only two immediate consequences: to bounding the size of the intersection of orbits of polynomial dynamical systems with affine subspaces and to the Waring problem over affine subspaces.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. E. Ben-Sasson and S. Kopparty, Affine dispersers from subspace polynomials, SIAM J. Comput. 41 (2012), 880–914.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Bourgain, Mordell’s exponential sum estimate revisited, J. Amer. Math. Soc. 18 (2005), 477–499.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Bourgain, On exponential sums in finite fields, in An Irregular Mind, János Bolyai Mathematical Society, Budapest, 2010, pp. 219–242.

    Chapter  Google Scholar 

  4. J. Bourgain and A. Glibichuk, Exponential sum estimates over a subgroup in an arbitrary finite field, J. Anal. Math. 115 (2011), 51–70.

    Article  MathSciNet  MATH  Google Scholar 

  5. M.-C. Chang, Polynomial iteration in characteristic p, J. Funct. Anal. 263 (2012), 3412–3421.

    Article  MathSciNet  MATH  Google Scholar 

  6. M.-C. Chang, Expansions of quadratic maps in prime fields, Proc. Amer. Math. Soc. 142 (2014), 85–92.

    Article  MathSciNet  MATH  Google Scholar 

  7. M.-C. Chang, Sparsity of the intersection of polynomial images of an interval, Acta Arith. 165 (2014), 243–249.

    Article  MathSciNet  MATH  Google Scholar 

  8. M.-C. Chang, J. Cilleruelo, M. Z. Garaev, J. Hernández, I. E. Shparlinski and A. Zumalacárregui, Points on curves in small boxes and applications, Michigan Math. J. 63 (2014), 503–534.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Cilleruelo, M. Z. Garaev, A. Ostafe and I. E. Shparlinski, On the concentration of points of polynomial maps and applications, Math. Z 272 (2012), 825–837.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Cilleruelo and I. E. Shparlinski, Concentration of points on curves in finite fields, Monatsh. Math. 171 (2013), 315–327.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Cipra, Waring’s number in a finite field, Integers 9 (2009), 435–440.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Cipra, T. Cochrane and C. G. Pinner, Heilbronn’s Conjecture on Waring’s number mod p, J. Number Theory 125 (2007), 289–297.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. Cochrane and C. Pinner, Sum-product estimates applied to Waring’s problem mod p, Integers 8 (2008), A46, 1–18.

    MathSciNet  MATH  Google Scholar 

  14. H. Delange, Sur les fonctions q-additives ou q-multiplicatives, Acta Arith. 21 (1972), 285–298.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Drmota, ‘The joint distribution of q-additive functions, Acta Arith. 100 (2001), 17–39.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. 13 (1967/1968), 259–265.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. J. Grabner, Completely q-multiplicative functions: the Mellin transform approach, Acta Arith. 65 (1993), 85–96.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Gutierrez and I. E. Shparlinski, Expansion of orbits of some dynamical systems over finite fields, Bull. Aust. Math. Soc. 82 (2010), 232–239.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Hofer, G. Larcher and F. Pillichshammer, Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions, Monatsh. Math. 154 (2008), 99–230.

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Kátai and M. V. Subbarao, Distribution of additive and q-additive functions under some conditions, Publ. Math. Debrecen 64 (2004), 167–187.

    MathSciNet  MATH  Google Scholar 

  21. R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  22. O. Roche-Newton and I. E. Shparlinski, Polynomial values in subfields and affine subspaces of finite fields, Quart. J. Math. 66 (2015), 693–706.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. H. Silverman and B. Viray, On a uniform bound for the number of exceptional linear subvarieties in the dynamical Mordell-Lang conjecture, Math. Res. Letters. 20 (2013), 547–566.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Weyl, Über die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77 (1916), 313–352.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Winterhof and C. van de Woestijne, Exact solutions to Waring’s problem in finite fields, Acta Arith. 141 (2010), 171–190.

    Article  MathSciNet  MATH  Google Scholar 

  26. T. D. Wooley, Vinogradov’s mean value theorem via efficient congruencing, II, Duke Math. J. 162 (2013), 673–730.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. D. Wooley, Translation invariance, exponential sums, and Waring’s problem, in Proceedings of the International Congress of Mathematicians, Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 505–529.

    Google Scholar 

  28. T. D. Wooley, Multigrade efficient congruencing and Vinogradovos mean value theorem, Proc. London Math. Soc. 111 (2015), 519–560.

    Article  MathSciNet  MATH  Google Scholar 

  29. B. Wu and Z. Liu, Linearized polynomials over finite fields revisited, Finite Fields Appl. 22 (2013), 79–100.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Igor Shparlinski for suggesting some extensions of initial results, and for his important comments on earlier versions of the paper. The author is also grateful to the Max Planck Institute for Mathematics for hosting the author for two months during the program “Dynamics and Numbers” when important progress on this paper was made. Finally, the author is grateful to the anonymous referee for helpful comments which improved the exposition.

During the preparation of this paper the author was supported by the UNSW Vice Chancellor’s Fellowship.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Alina Ostafe

Authors
  1. Alina Ostafe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alina Ostafe.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ostafe, A. Polynomial values in affine subspaces of finite fields. JAMA 138, 49–81 (2019). https://doi.org/10.1007/s11854-019-0021-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-019-0021-y

Search

Navigation