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Numerical semigroups generated by squares and cubes of three consecutive integers. (English) Zbl 1528.20100

Nathanson, Melvyn B. (ed.), Combinatorial and additive number theory III. Papers based on talks given at the CANT 2017 and 2018 workshops, New York, NY, USA, May 2017 and May 2018. Cham: Springer. Springer Proc. Math. Stat. 297, 101-121 (2020).
Summary: We derive the polynomial representations for minimal relations of the generating set of numerical semigroups \(R_n^k=\langle (n-1)^k,n^k,(n+1)^k\rangle \), \(k=2,3\). We find also the polynomial representations for degrees of syzygies in the Hilbert series \(H\left( z,R_n^k\right)\) of these semigroups, their Frobenius numbers \(F\left( R_n^k\right)\) and genera \(G\left( R_n^k\right) \). We discuss an extension of polynomial representations for minimal relations on numerical semigroups \(R_n^k\), \(k\ge 4\).
For the entire collection see [Zbl 1431.11004].

MSC:

20M14 Commutative semigroups
11D07 The Frobenius problem
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References:

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