Cyclotomic Numerical Semigroups

Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the unit disc. We conjecture that $S$ is a cyclotomic numerical semigroup if and only if $S$ is a complete intersection numerical semigroup and present some evidence for it. Aside from the notion of cyclotomic numerical semigroups we introduce the notion of cyclotomic exponents and polynomially related numerical semigroups. We derive some properties and give some applications of these new concepts.

[1]  S. Lang.  Algebraic Number Theory , 1971 .

[2]  J. C. Rosales,et al.  Families of numerical semigroups closed under finite intersections and adjoin of the Frobenius number , 2009 .

[4]  Micah J. Leamer,et al.  Huneke-Wiegand Conjecture for Complete Intersection Numerical Semigroup Rings , 2012, 1211.4554.

[5]  C. Delorme,et al.  Sous-monoïdes d’intersection complète de $N$ , 1976 .

[6]  László A. Székely,et al.  Generating functions for the Frobenius Problem with 2 and 3 generators , 1986 .

[7]  Steven H. Weintraub,et al.  Several Proofs of the Irreducibility of the Cyclotomic Polynomials , 2013, Am. Math. Mon..

[8]  T. Y. Lam,et al.  On the Cyclotomic Polynomial Φpq (X) , 1996 .

[9]  Manuel Delgado,et al.  On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators , 2013, IEEE Transactions on Information Theory.

[10]  Ernst Kunz,et al.  Die Wertehalbgruppe eines lokalen Rings der Dimension 1 , 1971 .

[11]  Jürgen Herzog,et al.  Generators and relations of abelian semigroups and semigroup rings , 1970 .

[12]  James H. Davenport,et al.  Effective Tests for Cyclotonic Polynomials , 1988, ISSAC.

[13]  I. Ojeda,et al.  Frobenius vectors, Hilbert series and gluings , 2013, 1311.1988.

[14]  Étienne Fouvry,et al.  On binary cyclotomic polynomials , 2013 .

[15]  Shaozu Wang.  Maximal height of divisors of $x^{pq^{b}}-1$ , 2013, 1312.3108.

[16]  J Bertin,et al.  Semi-groupes d'entiers et application aux branches , 1977 .

[17]  Ruud Pellikaan,et al.  The minimum distance of codes in an array coming from telescopic semigroups , 1995, IEEE Trans. Inf. Theory.

[18]  Pieter Moree,et al.  Numerical Semigroups, Cyclotomic Polynomials, and Bernoulli Numbers , 2013, Am. Math. Mon..

[19]  Pantelis A. Damianou,et al.  Monic Polynomials in Z[x] with Roots in the Unit Disc , 2001, Am. Math. Mon..

[20]  Nathan C. Ryan,et al.  Maximal height of divisors of $x\sp n-1$ , 2007 .

[21]  Bounds for the maximal height of divisors of xn−1 , 2009 .

[22]  Pieter Moree,et al.  Approximation of singular series and automata , 2000 .

[23]  M. Sahin,et al.  On free resolutions of some semigroup rings , 2012, 1211.4747.

[24]  Bartlomiej Bzdega.  Sparse binary cyclotomic polynomials , 2011, 1101.3720.