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arXiv · 0912.2513

A Diophantine Frobenius problem related to Riemann surfaces

Cormac O'Sullivan·Anthony Weaver
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Abstract

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 realized as a 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 ($t \geq 3$) is $NP$-hard, and it may be that our restricted problem retains this property.

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Cormac O'Sullivan, Anthony Weaver. 2009-12-13. A Diophantine Frobenius problem related to Riemann surfaces. https://arxiv.org/abs/0912.2513

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