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Analytic representations in the three-dimensional Frobenius problem

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Functional Analysis and Other Mathematics

Abstract

We consider the Diophantine problem of Frobenius for the semigroup \(\mathsf{S}(\mathbf{d}^{3})\) , where d 3 denotes the triple (d 1,d 2,d 3), gcd (d 1,d 2,d 3)=1. Based on the Hadamard product of analytic functions, we find the analytic representation of the diagonal elements a kk (d 3) of Johnson’s matrix of minimal relations in terms of d 1, d 2, and d 3. With our recent results, this gives the analytic representation of the Frobenius number F(d 3), genus G(d 3), and Hilbert series H(d 3;z) for the semigroups \(\mathsf{S}(\mathbf{d}^{3})\) . This representation complements Curtis’s theorem on the nonalgebraic representation of the Frobenius number F(d 3). We also give a procedure for calculating the diagonal and off-diagonal elements of Johnson’s matrix.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Technion, Haifa, 3200, Israel

    Leonid G. Fel

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  1. Leonid G. Fel
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Correspondence to Leonid G. Fel.

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Fel, L.G. Analytic representations in the three-dimensional Frobenius problem. Funct. Anal. Other Math. 2, 27–44 (2008). https://doi.org/10.1007/s11853-008-0014-3

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  • DOI: https://doi.org/10.1007/s11853-008-0014-3

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