Width bounds for Betti counts of four-generated numerical semigroups
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For a minimally four-generated numerical semigroup S=⟨n₀<n₁<n₂<n₃⟩, can its toric ideal always be generated by at most 2W elements, where W=(n₃−n₀)/gcd(n₁−n₀,n₂−n₀,n₃−n₀)? The target is over every field.
Why it matters and what counts as progress
Why it matters: A linear normalized-width bound improves quadratic relation-count estimates for every minimally four-generated numerical semigroup.
What counts as progress: Replace quadratic generator-count bounds by a uniform linear width bound or determine sharp extremal families.
How the problem developed
- 26 Jul 2024 · Related result [2]
Giulio Caviglia, Alessio Moscariello and Alessio Sammartano Proved the Herzog–Stamate width bound for four-generated numerical semigroups except at finitely many widths; their theorem covers all widths at least 40.
Ben Keene, GPT-6 Astra and Claude agents Reported a linear bound of at most 2W for the first Betti count of every minimally four-generated numerical semigroup, where W is the normalized width; Lean proves the upper-bound statements.
AI contribution
AI-led proof gives a linear Betti bound for four-generated semigroups
For a minimally four-generated numerical semigroup S=⟨n₀<n₁<n₂<n₃⟩, the minimal number of generators of its toric ideal is at most 2W over any field, where W=(n₃−n₀)/gcd(n₁−n₀,n₂−n₀,n₃−n₀). Consequently it is at most twice the ordinary width n₃−n₀.
Sources
- Herzog and Stamate, On the defining equations of the tangent cone of a numerical semigroup ring
- Earlier four-generator width result
- Keene, Linear upper bounds for the first Betti number of four-generated numerical semigroups: formalization and provenance
- Keene’s semigroup formalization
- Semigroup theorem correspondence and build record
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Publication dates and editorial changes.
AI-led proof gives a linear Betti bound for four-generated semigroups