Fibonacci-like growth of numerical semigroups of a given genus |
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Authors: | Alex Zhai |
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Affiliation: | 1. 1 Saint Francis Place Apt. 1508, San Francisco, CA, 94107, USA
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Abstract: | We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if n g is the number of numerical semigroups of genus g, we prove that $$lim_{g rightarrow infty} n_g varphi^{-g} = S $$ where $varphi = frac{1 + sqrt{5}}{2}$ is the golden ratio and S is a constant, resolving several related conjectures concerning the growth of n g . In addition, we show that the proportion of numerical semigroups of genus g satisfying f<3m approaches 1 as g→∞, where m is the multiplicity and f is the Frobenius number. |
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