Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913951620530176 |
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| author | Bogart, Tristram Morales, Santiago |
| author_facet | Bogart, Tristram Morales, Santiago |
| contents | De Loera, O'Neill and Wilburne introduced a general model for random numerical semigroups in which each positive integer is chosen independently with some probability p to be a generator, and proved upper and lower bounds on the expected Frobenius number and expected embedding dimensions. We use a range of probabilistic methods to improve the upper bounds to within a polylogarithmic factor of the lower bounds in each case. As one of the tools to do this, we prove that for any prime q, if A is a random subset of the cyclic group Z_q whose size is of order log(q) and k is also of order log(q), then with high probability the k-fold sumset kA is all of Z_q. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups Bogart, Tristram Morales, Santiago Commutative Algebra Combinatorics Number Theory 20M14, 06F05, 05D40, 11P70 De Loera, O'Neill and Wilburne introduced a general model for random numerical semigroups in which each positive integer is chosen independently with some probability p to be a generator, and proved upper and lower bounds on the expected Frobenius number and expected embedding dimensions. We use a range of probabilistic methods to improve the upper bounds to within a polylogarithmic factor of the lower bounds in each case. As one of the tools to do this, we prove that for any prime q, if A is a random subset of the cyclic group Z_q whose size is of order log(q) and k is also of order log(q), then with high probability the k-fold sumset kA is all of Z_q. |
| title | Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups |
| topic | Commutative Algebra Combinatorics Number Theory 20M14, 06F05, 05D40, 11P70 |
| url | https://arxiv.org/abs/2411.13767 |