Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T06:35:34.915Z Has data issue: false hasContentIssue false
  • English
  • Français

A DIOPHANTINE FROBENIUS PROBLEM RELATED TO RIEMANN SURFACES

Published online by Cambridge University Press:  10 March 2011

CORMAC O'SULLIVAN  and
ANTHONY WEAVER
CORMAC O'SULLIVAN
Affiliation:
Department of Mathematics and Computer Science, Bronx Community College, City University of New York, 2155 University Avenue, Bronx, New York 10453, USA e-mails: cormac.osullivan@bcc.cuny.edu; anthony.weaver@bcc.cuny.edu
ANTHONY WEAVER
Affiliation:
Department of Mathematics and Computer Science, Bronx Community College, City University of New York, 2155 University Avenue, Bronx, New York 10453, USA e-mails: cormac.osullivan@bcc.cuny.edu; anthony.weaver@bcc.cuny.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair (p, q), 2 < p < q, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realised as semi-regular pq-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order pq. The general t-dimensional Frobenius problem (t ≥ 3) is NP-hard, and it may be that our restricted problem retains this property.

Keywords

Primary 14J5011D04
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 501 - 522
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Breuer, T., Characters and automorphism groups of compact Riemann surfaces (Cambridge University Press, Cambridge, UK, 2001).Google Scholar
2.Curtis, F., On formulas for the Frobenius number of a numerical semigroup, Math. Scand. 67 (1990), 190–192.CrossRefGoogle Scholar
3.Farkas, H. M. and Kra, I., Riemann surfaces, 2nd edn. (Graduate Texts in Mathematics 71) (Springer-Verlag, New York, NY, 1980).CrossRefGoogle Scholar
4.Greenberg, L., Finiteness theorems for Fuchsian and Kleinian groups, in Discrete groups and automorphic functions (Harvey, W. J., Editor) (Academic Press, London, 1977).Google Scholar
5.Hall, M., The theory of groups, 2nd edn. (Chelsea Publishing Company, New York, NY, 1976).Google Scholar
6.Kulkarni, R. S., Symmetries of surfaces, Topology 26 (1987) 195–203.CrossRefGoogle Scholar
7.Kulkarni, R. S. and Maclachlan, C., Cyclic p-groups of symmetries of surfaces, Glasgow Math. J. 33 (1991) 213–221.CrossRefGoogle Scholar
8.Maclachlan, C. and Miller, A., Generating functions for finite group actions on surfaces, Math. Proc. Camb. Phil. Soc. 124 (1) (1998) 21–49.CrossRefGoogle Scholar
9.Miranda, R., Algebraic curves and Riemann surfaces (Graduate Studies in Mathematics 5) (American Mathematical Society, Providence, RI, 1995).CrossRefGoogle Scholar
10.Ramirez-Alfonsin, J. L., Complexity of the Frobenius problem, Combinatorica 16 (1) (1996) 143–147.CrossRefGoogle Scholar
11.Ramirez-Alfonsin, J. L., The diophantine Frobenius problem (Oxford Lecture Series in Mathematics and its Applications 30) (Oxford University Press, Oxford, UK, 2005).CrossRefGoogle Scholar
12.Sylvester, J. J., Problem 7382, Educ. Times 37 (1884) 21.Google Scholar
13.Weaver, A., Genus spectra for split metacyclic groups, Glasgow Math J. 43 (2001) 209–218.CrossRefGoogle Scholar