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A new upper bound for the cross number of finite abelian groups

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Abstract

In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite abelian groups. Given a finite abelian group, this upper bound appears to depend only on the rank and the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite abelian groups holds asymptotically in at least two different directions.

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References

  1. N. Alon and M. Dubiner, A lattice point problem and additive number theory, Combinatorica 15 (1995), 301–309.

    Article  MathSciNet  Google Scholar 

  2. P. Baginski, S. T. Chapman, K. McDonald and L. Pudwell, On cross numbers of minimal zero sequences in certain cyclic groups, Ars Combinatoria 70 (2004), 47–60.

    MathSciNet  MATH  Google Scholar 

  3. S. T. Chapman, On the Davenport constant, the cross number, and their application in factorization theory, in Zero-dimensional Commutative Rings (Knoxville, TN, 1994), Lecture Notes in Pure and Applied Mathematics, Dekker, New-York, Vol. 171, 1995, pp. 167–190.

    MathSciNet  MATH  Google Scholar 

  4. S. T. Chapman and A. Geroldinger, On cross numbers of minimal zero sequences, Australasian Journal of Combinatorics 14 (1996), 85–92.

    MathSciNet  MATH  Google Scholar 

  5. Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin and L. Rackham, Zero-sum problems in finite abelian groups and affine caps, Quarterly Journal of Mathematics 58 (2007), 159–186.

    Article  MathSciNet  Google Scholar 

  6. S. Elledge and G. H. Hurlbert, An application of graph pebbling to zero-sum sequences in abelian groups, Integers 5 (2005), #A17.

    MathSciNet  MATH  Google Scholar 

  7. P. van Emde Boas, A combinatorial problem on finite abelian groups II, Reports ZW-1969-007, Math. Centre, Amsterdam, 1969.

    Google Scholar 

  8. W. Gao and A. Geroldinger, Systems of sets of lengths II, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 70 (2000), 31–49.

    Article  MathSciNet  Google Scholar 

  9. W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey, Expositiones Mathematicae 24 (2006), 337–369.

    Article  MathSciNet  Google Scholar 

  10. W. Gao, Q. H. Hou, W. A. Schmid and R. Thangadurai, On short zero-sum subsequences II, Integers 7 (2007), #A21.

    MathSciNet  MATH  Google Scholar 

  11. A. Geroldinger, On a conjecture of Kleitman and Lemke, Journal of Number Theory 44 (1993), 60–65.

    Article  MathSciNet  Google Scholar 

  12. A. Geroldinger, The cross number of finite abelian groups, Journal of Number Theory 48 (1994), 219–223.

    Article  MathSciNet  Google Scholar 

  13. A. Geroldinger and F. Halter-Koch, Non-unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, Vol. 278, Chapman & Hall/CRC, 2006.

  14. A. Geroldinger and F. Halter-Koch, Non-unique factorizations: a survey, in Multiplicative Ideal Theory in Commutative Algebra, Springer, New York, 2006, pp. 207–226.

    Book  Google Scholar 

  15. A. Geroldinger and Y. Hamidoune, Zero-sumfree sequences in cyclic groups and some arithmetical application, Journal de Théorie des Nombres de Bordeaux 14 (2002), 221–239.

    Article  MathSciNet  Google Scholar 

  16. A. Geroldinger and R. Schneider, The cross number of finite abelian groups II, European Journal of Combinatorics 15 (1994), 399–405.

    Article  MathSciNet  Google Scholar 

  17. A. Geroldinger and R. Schneider, The cross number of finite abelian groups III, Discrete Mathematics 150 (1996), 123–130.

    Article  MathSciNet  Google Scholar 

  18. A. Geroldinger and R. Schneider, On minimal zero sequences with large cross number, Ars Combinatoria 46 (1997), 297–303.

    MathSciNet  MATH  Google Scholar 

  19. U. Krause, A characterization of algebraic number fields with cyclic class group of prime power order, Mathematische Zeitschrift 186 (1984), 143–148.

    Article  MathSciNet  Google Scholar 

  20. U. Krause and C. Zahlten, Arithmetic in Krull monoids and the cross number of divisor class groups, Mitteilungen der Mathematischen Gesellschaft in Hamburg, Vol. 12, 1991, pp. 681–696.

    MathSciNet  MATH  Google Scholar 

  21. J. E. Olson, A combinatorial problem on finite abelian groups I, Journal of Number Theory 1 (1969), 8–10.

    Article  MathSciNet  Google Scholar 

  22. J. E. Olson, A combinatorial problem on finite abelian groups II, Journal of Number Theory 1 (1969), 195–199.

    Article  MathSciNet  Google Scholar 

  23. A. Plagne and W. A. Schmid, On large half-factorial sets in elementary p-groups: maximal cardinality and structural characterization, Israel Journal of Mathematics 145 (2005), 285–310.

    Article  MathSciNet  Google Scholar 

  24. A. Plagne and W. A. Schmid, On the maximal cardinality of half-factorial sets in cyclic groups, Mathematische Annalen 333 (2005), 759–785.

    Article  MathSciNet  Google Scholar 

  25. M. Radziejewski and W. A. Schmid, Weakly half-factorial sets in finite abelian groups, Forum Mathematicum 19 (2007), 727–747.

    Article  MathSciNet  Google Scholar 

  26. J.B. Rosser, The n-th prime is greater than n log n, Proceedings of the London Mathematical Society (2) 45 (1939), 21–44.

    Article  MathSciNet  Google Scholar 

  27. P. Samuel, Théorie Algébrique des Nombres, Hermann, 2003.

  28. W. A. Schmid, Differences in sets of lengths of Krull monoids with finite class group, Journal de Théorie des Nombres de Bordeaux 17 (2005), 323–345.

    Article  MathSciNet  Google Scholar 

  29. A. Schrijver, Theory of Linear and Integer Programming, Wiley, 1998.

  30. G. Tenenbaum, Introduction à la Théorie Analytique et Probabiliste des Nombres, Cours spécialisés, SMF, 1995.

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  1. Centre de Mathématiques Laurent Schwartz, UMR 7640 du CNRS École polytechnique, 91128, Palaiseau Cedex, France

    Benjamin Girard

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  1. Benjamin Girard
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Correspondence to Benjamin Girard.

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Girard, B. A new upper bound for the cross number of finite abelian groups. Isr. J. Math. 172, 253–278 (2009). https://doi.org/10.1007/s11856-009-0072-3

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  • DOI: https://doi.org/10.1007/s11856-009-0072-3

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