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Cyclotomic numerical semigroups. (English) Zbl 1343.20060

A numerical semigroup \(S\) (i.e., a sub-semigroup of the additive semigroup of positive integers, containing all sufficiently large integers) is called a cyclotomic semigroup if its polynomial \[ P_S(X)=1+(X-1)\sum_{s\not\in S}X^s \] is a product of cyclotomic polynomials. The authors conjecture that the family of cyclotomic semigroups coincides with the family of complete intersection semigroups, note that this is true for all semigroups with Frobenius number \(\leq 70\), as well as for all numerical semigroups having a minimal generating subset of at most three elements. They show also that for every numerical semigroup \(S\) one has \[ P_S(X)=\prod_{j=1}^\infty(1-X^j)^{e_j} \] with integral exponents \(e_j\) (cyclotomic exponents), and explore properties of these exponents. The authors call two numerical semigroups \(S,T\) polynomially related if there exists a polynomial \(f\in Z[X]\) and an integer \(w\geq 1\) such that \[ P_S(X^w)f(X)=P_T(X){X^w-1\over X-1}, \] and give some applications of this notion.

MSC:

20M14 Commutative semigroups
11C08 Polynomials in number theory
11D07 The Frobenius problem
11B83 Special sequences and polynomials

Software:

numericalsgps
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Full Text: DOI arXiv

References:

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