On the smallest numerical semigroups closed under affine maps

Abstract

We study numerical semigroups $S_{a,b}(m)$ generated by the orbit of $m$ under the affine map $T_{a,b}(z)=az+b$, where $a\ge 2$, $m>1$, $\gcd(b,m)=1$, and $b\ge -(a-2)m-2$. This extends the usual affine-closed setting to feasible negative values of $b$. We write $A_i=(a^i-1)/(a-1)$ and let $n$ be the smallest positive integer such that $A_n\ge m$. We determine the minimal generators and give an explicit description of the Apéry set, obtaining homogeneity and formulas for the Frobenius number and the genus. We also study pseudo-Frobenius numbers via the induced Apéry parametrization and prove the sharp upper bound $\operatorname{t}(S_{a,b}(m))\le n-1$ for the type. We give a complete characterization of the symmetric members of the family in terms of the canonical representative of $m-1$. Finally, we exhibit a subfamily whose pseudo-Frobenius numbers form an arithmetic progression of length $n-1$; in particular, this subfamily attains the bound.

Disclosure

“to adapt the Gröbner basis and saturation arguments used in [2] for the generalized repunit case. This question, together with a detailed Gröbner basis analysis of I2 (Xc ), will be ad- dressed in a subsequent work. Acknowledgements. Microsoft Copilot was used to assist with English language editing and stylistic revision. The authors take full responsibility for the final content of the manuscript. References [1] Barucci, V., Dobbs, D. E. an”

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2026-08-24T00:28:51.963Z

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