On the enumeration of Arf numerical semigroups with given multiplicity and conductor

Abstract: The number of numerical semigroups with given Frobenious number (or conductor, or genus) is one of the topics that is studied by many researchers. In our previous works, we have given parametrizations of Arf numerical semigroups of small multiplicity and obtained formulas for the number of Arf numerical semigroups with multiplicity less than 14 and arbitrary conductor. I presented part of these results in ODTÜ-Bikent AG seminars 6 years ago. We noticed that the number of Arf numerical semigroups with multiplicity $m$ and conductor $c$ is (eventually) constant for some $m$ (especially for prime $m$) when restricted to some congruence classes of $c$ modulo $m$. In a recent work with N. Tutaş, we have characterized those multiplicities $m$ and congruence classes of $c$ modulo $m$ for which the above property holds. This talk will be based on [Karakaş H İ and Tutaş N, (2025), On the enumeration of Arf numerical semigroups with given multiplicity and conductor, Semigroup Forum 110, 308-316.] where the above characterization is given.

algebraic geometry

Audience: researchers in the discipline

ODTU-Bilkent Algebraic Geometry Seminars