Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908759735926784 |
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| author | Morales, Marcel Dung, Nguyen Thi |
| author_facet | Morales, Marcel Dung, Nguyen Thi |
| contents | Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a, ha + d, ha + 2d, . . . , ha + kd, c, that is its embedding dimension is k + 2. In a previous work, the authors described the Ap{é}ry set and a Gr{ö}bner basis of the ideal defining S under one technical assumption, the complete version will be published in a forthcoming paper. In this paper we continue with this assumption and we describe the Pseudo Frobenius set. As a consequence we give a complete description of S when it is symmetric or almost symmetric as well as generalize and extend the previous results of Ignacio Garc{í}a-Marco, J. L. Ram{í}rez Alfons{í}n and O. J. R{ø}dseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. Garc{í}a-S{á}nchez. Moreover, for given numbers a, d, k, h, c, a simple algorithm allows us to determine if S is almost symmetric or not and furthermore to find its type and Frobenius number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07467 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups Morales, Marcel Dung, Nguyen Thi Commutative Algebra Algebraic Geometry Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a, ha + d, ha + 2d, . . . , ha + kd, c, that is its embedding dimension is k + 2. In a previous work, the authors described the Ap{é}ry set and a Gr{ö}bner basis of the ideal defining S under one technical assumption, the complete version will be published in a forthcoming paper. In this paper we continue with this assumption and we describe the Pseudo Frobenius set. As a consequence we give a complete description of S when it is symmetric or almost symmetric as well as generalize and extend the previous results of Ignacio Garc{í}a-Marco, J. L. Ram{í}rez Alfons{í}n and O. J. R{ø}dseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. Garc{í}a-S{á}nchez. Moreover, for given numbers a, d, k, h, c, a simple algorithm allows us to determine if S is almost symmetric or not and furthermore to find its type and Frobenius number. |
| title | Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2601.07467 |