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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
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Abstract

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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

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