Skip to main content
SpringerLink
Log in

On the directions problem in AG(n, q)

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

We prove that if q = p h, p a prime, do not exist sets \({U {\subseteq} AG(n,q)}\), with |U| = q k and 1 < k < n, determining N directions where

$$ \frac{{q^k} - 1}{p - 1} < N \le \frac{q+3}{2} q ^{k-1}+ q^{k-2} +\dots+q{^2} + q $$

when q is odd and

$$ \frac{{q^k} - 1}{3} < N \le \frac{q+3}{2} q ^{k-1}+ q^{k-2} +\dots+q{^2} + q $$

when q > 2 is even.

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. Ball S.: The number of directions determined by a function over a finite field. J. Comb. Theory. Ser. A 104, 341–350 (2003)

    Article  MATH  Google Scholar 

  2. Blokhuis A., Ball S., Brouwer A.E., Storme L., Szönyi T.: On the number of slopes of the graph of a function defined on a finite field. J. Comb. Theory. Ser. A 86, 187–196 (1999)

    Article  MATH  Google Scholar 

  3. Gács, A.: Polynomials for higher dimensional problems, Combinatorics 2008 (to appear in Discrete Math.)

  4. Gács, A., Lovász, L., Szönyi, T.: Directions in AG(2, p 2). Innov. Inc. Geom. 6–7, 189–201 (2007–2008)

  5. Hirschfeld J.W.P.: Projective geometries over finite fields, 2nd edn. Oxford Science Publication, Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  6. Lunardon G.: Linear k-blocking sets. Combinatorica 21, 571–581 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Rédei, L.: Luckenhafte Polynome uber endlichen Korpen, Birkhauser-Verlag, Basel (1970) (English translation: Lacunary polynomials over finite fields), North-Holland, Amsterdam (1973)

  8. Storme L., Sziklai P.: Linear point sets and Rédei type k-blocking sets in PG(n, q). J Algebr Combinatorics 14, 221–228 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso di Monte S. Angelo, Edificio T, via Cintia, 80134, Naples, Italy

    Paola De Vito

Authors
  1. Paola De Vito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paola De Vito.

Additional information

Communicated by Editor in Chief.

Rights and permissions

Reprints and permissions

About this article

Cite this article

De Vito, P. On the directions problem in AG(n, q). Ricerche mat. 60, 39–43 (2011). https://doi.org/10.1007/s11587-010-0094-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-010-0094-5

Keywords

Search

Navigation