Totally positive elements with $m$ partitions exist in almost all real quadratic fields
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914144817512448 |
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| author | Zindulka, Mikuláš |
| author_facet | Zindulka, Mikuláš |
| contents | In this paper, we study partitions of totally positive integral elements $α$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper bound for the norm of $α$ that can be represented as a sum of indecomposables in at most $m$ ways, completely characterize the $α$'s represented in exactly $2$ ways, and subsequently apply this result to complete the search for fields containing an element with $m$ partitions for $1 \leq m \leq 7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18080 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Totally positive elements with $m$ partitions exist in almost all real quadratic fields Zindulka, Mikuláš Number Theory 11R11 (Primary) 11R80, 11A55, 11P81 (Secondary) In this paper, we study partitions of totally positive integral elements $α$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper bound for the norm of $α$ that can be represented as a sum of indecomposables in at most $m$ ways, completely characterize the $α$'s represented in exactly $2$ ways, and subsequently apply this result to complete the search for fields containing an element with $m$ partitions for $1 \leq m \leq 7$. |
| title | Totally positive elements with $m$ partitions exist in almost all real quadratic fields |
| topic | Number Theory 11R11 (Primary) 11R80, 11A55, 11P81 (Secondary) |
| url | https://arxiv.org/abs/2409.18080 |