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Herzog–Stamate width conjecture on minimal generators of numerical semigroup defining ideals

Prove that for every numerical semigroup S minimally generated by positive integers s_1 < s_2 < ... < s_g with width wd(S) = s_g − s_1, the minimal number of generators μ(I_S) of the binomial defining ideal I_S of the semigroup ring k[S] satisfies μ(I_S) ≤ (wd(S)+1 choose 2).

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Background

Herzog and Stamate studied Sally semigroups and observed tight bounds on generators of the defining ideals of associated graded rings. Motivated by these observations, they proposed a general bound depending only on the width of the numerical semigroup.

This paper introduces "Sally type" numerical semigroups and proves exact formulas for their Frobenius numbers and minimal numbers of generators. It also verifies that Sally-type semigroups satisfy the width conjecture, lending evidence in a broad but still special class.

References

They showed that $S=\langle e,e+1,e+4,\hdots,2e-1 \rangle$ is a symmetric Sally semigroup and the number of minimal generators $\mu(I\ast_S)$ of $I\ast_S$ satisfies \mu(I*_S)={e-2 \choose 2}\leq {wd(S)+1 \choose 2}. and conjectured that $\displaystyle\mu(I_S)\leq {wd(S)+1 \choose 2}$.

— Numerical Semigroups of Sally Type (2507.11738 - Dubey et al., 15 Jul 2025) in Section 1, Introduction