136 research outputs found
Cyclotomic numerical semigroups
Given a numerical semigroup , we let be its semigroup polynomial. We study cyclotomic numerical semigroups;
these are numerical semigroups such that has all its roots
in the unit disc. We conjecture that is a cyclotomic numerical semigroup if
and only if is a complete intersection numerical semigroup and present some
evidence for it. Aside from the notion of cyclotomic numerical semigroup we
introduce the notion of cyclotomic exponents and polynomially related numerical
semigroups. We derive some properties and give some applications of these new
concepts.Comment: 17 pages, accepted for publication in SIAM J. Discrete Mat
Cyclotomic exponent sequences of numerical semigroups
We study the cyclotomic exponent sequence of a numerical semigroup and we compute its values at the gaps of the elements of with unique representations in terms of minimal generators, and the Betti elements for which the set is totally ordered with respect to (we write whenever with ). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, García-Sánchez and Moree (2016) stating that is cyclotomic if and only if it is a complete intersection
Cyclotomic coefficients: gaps and jumps
We improve several recent results by Hong, Lee, Lee and Park (2012) on gaps
and Bzd\c{e}ga (2014) on jumps amongst the coefficients of cyclotomic
polynomials. Besides direct improvements, we also introduce several new
techniques that have never been used in this area.Comment: 25 page
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