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Frobenius problem and the covering radius of a lattice

Abstract

Let N2N \geq2 and let 1<a1<...<aN1 < a_1 < ... < a_N be relatively prime integers. Frobenius number of this NN-tuple is defined to be the largest positive integer that cannot be expressed as i=1Naixi\sum_{i=1}^N a_i x_i where x1,...,xNx_1,...,x_N are non-negative integers. The condition that gcd(a1,...,aN)=1gcd(a_1,...,a_N)=1 implies that such number exists. The general problem of determining the Frobenius number given NN and a1,...,aNa_1,...,a_N is NP-hard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating Frobenius number to the covering radius of the null-lattice of this NN-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.Comment: 12 pages; minor revisions; to appear in Discrete and Computational Geometr

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