Cyclotomic numerical semigroups. (Q2800181)

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 NEWLINE\[NEWLINEP_S(X)=1+(X-1)\sum_{s\not\in S}X^sNEWLINE\]NEWLINE 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 NEWLINE\[NEWLINEP_S(X)=\prod_{j=1}^\infty(1-X^j)^{e_j}NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINEP_S(X^w)f(X)=P_T(X){X^w-1\over X-1},NEWLINE\]NEWLINE and give some applications of this notion.