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Moscariello–Sammartano bounded-type problem for numerical semigroups

Does there exist a function f:N→Nf:\mathbb{N}\to\mathbb{N} such that, for every numerical semigroup SS with embedding dimension 44, the type satisfies t(S)≤f(η(S))t(S)\le f\bigl(\eta(S)\bigr), where η(S)\eta(S) is the cardinality of a minimal presentation?

References

Primary source

arXiv

Additional references

commutative-algebranumerical-semigroups
Posted

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims to settle this previously open question for four-generated numerical semigroups, but the result has not been independently verified.

The 2024 paper Open problems on relations of numerical semigroups formulated the question as Problem 25: for embedding dimension 44, is the type t(Γ)t(\Gamma) bounded by a function of the cardinality ρ(Γ)\rho(\Gamma) of a minimal presentation? It recorded the question as open.

Known results

Earlier work established the converse-type estimate ρ(Γ)≤4+9t(Γ)\rho(\Gamma)\leq 4+9t(\Gamma), cited in the 2024 paper as [B88, Theorem 7].

September 2026 claimed resolution

The newly posted preprint The type and cardinality of minimal presentations of numerical semigroups with embedding dimension four claims new bounds relating t(S)t(S) and η(S)\eta(S), thereby resolving the boundedness question and improving the previous upper bound. This claim is unrefereed and remains unverified.

Current status (as of September 2026): A September 2026 preprint claims the 44-generated case is solved with sharper bounds, but verification is outstanding.

Sources

Solutions 0

No solutions have been posted yet.