Skip to main content
Log in

On the largest degree of an irreducible factor of a polynomial in F q [X]

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

References

  1. M. Car, Factorization dansF q [X],CR Acad. Sci. Paris, Ser. I,294, 147–150 (1982).

    MATH  MathSciNet  Google Scholar 

  2. M. Car, Théorèmes de densité dansF q [X],Acta Arith.,48, 145–165 (1987).

    MATH  MathSciNet  Google Scholar 

  3. P. Erdős and P. Turán, On some problems of a statistical grouptheory I,Z. Wahr. verw. Geb.,4, 175–186 (1965).

    Article  Google Scholar 

  4. P. Flajolet and A. Odlyzko, Singularity analysis of generating functions,SIAM J. Discrete Math.,3, 216–240 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Flajolet and M. Soria, Gaussian limiting distributions for the number of components in combinatorial structures,J. Combinatorial Theory, Ser. A,53, 165–182 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  6. V. L. Goncharov, Sur la distribution des cycles dans les permutations,Dokl. Akad. Nauk SSSR,35, 267–269 (1942).

    MATH  Google Scholar 

  7. K.-H. Indlekofer and E. Manstavičius, Additive and multiplicative functions on arithmetical semigroups,Publ. Math. Debrecen,45, 1–17 (1994).

    MATH  MathSciNet  Google Scholar 

  8. A. Knopfmacher, On the number of distinct degree sizes of a polynomial over a finite field (1995) (to appear).

  9. A. Knopfmacher and J. Knopfmacher, Counting irreducible factors of polynomials over a finite field,Discrete Math.,112, 103–118 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Knopfmacher and J. N. Ridley, Reciprocal sums over partitions and compositions,SIAM J. Discrete Math.,6, 388–399 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Knopfmacher and R. Warlimont, Distinct degree factorizations for polynomials over a finite field,Trans. Amer. Math. Soc. 347, 2235–2243 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Knopfmacher and R. Warlimont, Counting permutations and polynomials with a restricted factorization pattern (to appear inAustralas. J. Combin.).

  13. J. Knopfmacher, Direct factors of polynomial rings over finite fields,J. Combin. Theory, Ser. A,40, 429–434 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  14. V. F. Kolchin and V. P. Chistyakov, On the cycle structure of random permutations,Mat. Zametki,18, 929–938 (1975).

    MATH  MathSciNet  Google Scholar 

  15. E. Manstavičius, Semigroup elements free of large prime factors, in:New Trends in Probability and Statistics, vol. 2:Analytic and Probabilistic Methods in Number Theory, F. Schweiger and E. Manstavičius (Eds.), TEV, Vilnius/VSP, Utrecht (1992), pp. 135–153.

    Google Scholar 

  16. E. Manstavičius, Remarks on the semigroup elements free of large prime factors,Lith. Math. J.,32, 400–410 (1992).

    Article  Google Scholar 

  17. E. Manstavičius and R. Skrabutènas, Summation of the values of multiplicative functions semigroups,Lith. Math. J.,33, 330–340 (1993).

    Google Scholar 

  18. E. Manstavičius and R. Skrabutènas. Local distributions of additive functions on arithmetical semigroups. Preprint 95-11, Vilnius University, Faculty of Mathematics (1995).

  19. M. Mignotte and J. L. Nicolas, Statistique surF q [X],Ann. Inst. Henri Poincaré, Sec. B,19, 113–121 (1983).

    MATH  MathSciNet  Google Scholar 

  20. J. L. Nicolas, A Gaussian law onF q [X],Colloq Math. Soc. János Bolyai. Topics in Classical Number Theory,34, 1127–1162 (1981).

    MathSciNet  Google Scholar 

  21. J. L. Nicolas, Distribution statistique de l'ordre d'un element du groupe symetrique,Acta Math. Hungar.,45, 96–184 (1985).

    Article  MathSciNet  Google Scholar 

  22. A. M. Odlyzko, Discrete logarithms in finite fields and their cryptographic significance,Lecture Notes in Computer Science, Advances in Cryptology,209, 224–314 (1985).

    MATH  MathSciNet  Google Scholar 

  23. L. A. Shepp and S. P. Lloyd, Ordered cycle lengths in a random permutation,Trans. Amer. Math. Soc.,121, 340–357 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  24. R. Warlimont, Arithmetical semigroups, II: sieving by large and small prime elements, sets of multiples,Manuscripta Math.,71, 197–221 (1991).

    MATH  MathSciNet  Google Scholar 

  25. R. Warlimont, Arithmetical semigroups, IV: Selberg's analysis,Arch. Math. (Basel),60, 58–72 (1993).

    MATH  MathSciNet  Google Scholar 

  26. H. S. Wilf, Three problems in combinatorial asymptotics,J. Combin. Theory. Ser. A,35, 199–207 (1983).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The paper was written during the second author's visiting professorship in the Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand. Johannesburg.

Translated from Lietuvos Matematikos Rinkinys, Vol. 37, No. 1, pp. 50–60, January–March, 1997.

Translated by E. Manstavičius

Rights and permissions

Reprints and permissions

About this article

Cite this article

Knopfmacher, A., Manstavičius, E. On the largest degree of an irreducible factor of a polynomial in F q [X]. Lith Math J 37, 38–45 (1997). https://doi.org/10.1007/BF02465438

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02465438

Keywords

Navigation