Analytic representations in the three-dimensional Frobenius problem |
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Authors: | Leonid G Fel |
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Institution: | (1) Department of Civil and Environmental Engineering, Technion, Haifa, 3200, Israel |
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Abstract: | We consider the Diophantine problem of Frobenius for the semigroup
, 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
. 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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Keywords: | Semigroups Frobenius problem Hilbert series of a graded ring Hadamard product of analytic functions |
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