On Wilf's conjecture for numerical semigroups

Abstract: A numerical semigroup $S$ is a cofinite submonoid of $\mathbb{N}$. This means that $S$ is stable under addition, contains $0$, and has finite complement in $\mathbb{N}$. Some important numbers attached to $S$ are its genus $g = \text{card}(\mathbb{N} \setminus S)$, its conductor $c=\max(\mathbb{Z} \setminus S)+1$ and its minimal number $n$ of generators. Half a century ago, Herbert Wilf came up with a very clever conjectural upper bound on the genus $g$ in terms of $c$ and $n$, namely $$g \le c(1-1/n).$$ In this talk, we will provide a brief overview of the current status of Wilf's conjecture and discuss connections with additive combinatorics, graph theory and commutative algebra.

commutative algebracombinatoricsnumber theory

Audience: researchers in the discipline

New York Number Theory Seminar