Dice Question Streamline Icon: https://streamlinehq.com

Asymptotic order of the expected embedding dimension in the ER-type model

Determine the asymptotic order of the expected embedding dimension E[e((p))] for the unconstrained Erdős–Rényi-type random numerical semigroup model (p), specifically by proving that E[e((p))] = Θ(log(1/p)).

Information Square Streamline Icon: https://streamlinehq.com

Background

Parallel to the Frobenius number, the authors’ empirical paper indicates that the expected embedding dimension has logarithmic growth in 1/p for the unconstrained ER-type model.

This conjecture, if proven, would refine understanding of generator selection and minimality in random numerical semigroups and align upper bounds more closely with observed behavior.

References

However, extensive experiments , in which 1000 Erd\H{o}s-R enyi semigroups were generated for each of fifteen values of $p$, suggest the following conjecture.\n\nConjecture.\n\n2. The expected embedding dimension $\mathbb{E}[\e((p))]$ is of order $\log \left(\frac{1}{p}\right)$.

— Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups (2411.13767 - Bogart et al., 21 Nov 2024) in Section 5 (Experiments, Conclusions, and Future Work)