A DIOPHANTINE FROBENIUS PROBLEM RELATED TO RIEMANN SURFACES

CORMAC O'SULLIVAN, ANTHONY WEAVER
2011 Glasgow Mathematical Journal  
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
more » ... so 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. 2010 Mathematics Subject Classification. Primary 14J50, 11D04. Introduction. A set of integers {a 1 , a 2 , . . . a t }, t ≥ 2, with a i > 1 and gcd = 1, has a Frobenius number which is the largest positive integer not representable in the form k 1 a 1 + k 2 a 2 + · · · + k t a t , where each k i is a non-negative integer. It is a simple exercise to show that g({a 1 , a 2 , . . . , a t }) exists under the stated conditions. Finding g({a 1 , . . . , a t }) for a given set {a 1 , . . . , a t } is the linear Diophantine Frobenius problem [11] . In 1884, Sylvester [12] established the formula for the two-dimensional Frobenius number. In 1990, it was shown by Curtis [2] that for t ≥ 3 there is no finite set of polynomials {f 1 , . . . , f k } in t variables such that, for each t-tuple {a 1 , a 2 , . . . , a t } with greatest common divisor 1, g({a 1 , a 2 , . . . , a t }) = f i (a 1 , a 2 , . . . , a t ) for some i. Algorithms for computing the t-dimensional Frobenius numbers exist [11] , but the problem (for variable t ≥ 3) is NP-hard [10]. Throughout the paper, p, q will be primes satisfying 2 < p < q with p , q denoting the integers (p − 1)/2 and (q − 1)/2, respectively. The four integers d 0 = pq, d 1 = p q, d 2 = pq , d 3 = (pq − 1)/2 (1.2) have gcd = 1, so they determine the four-dimensional Frobenius number (1.3) https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s0017089511000097 fatcat:cd3lao5swfbapijiwj6oxipybe