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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.08387 |
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| _version_ | 1866912663529848832 |
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| author | Freitag, James |
| author_facet | Freitag, James |
| contents | We show that if any four distinct solutions of a rational difference equation are algebraically independent, then any number of distinct solutions to the equation are independent. A nontrivial variant of this result is given for autonomous difference equations or algebraic dynamical systems, where we show the degree of nonminimality is at most one. The results have a natural interpretation in terms of invariant or periodic subvarieties of algebraic dynamical systems and $σ$-varieties. Surprisingly, the proofs of these results rely on the classification of finite simple groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | When any four solutions are independent Freitag, James Logic Algebraic Geometry Dynamical Systems We show that if any four distinct solutions of a rational difference equation are algebraically independent, then any number of distinct solutions to the equation are independent. A nontrivial variant of this result is given for autonomous difference equations or algebraic dynamical systems, where we show the degree of nonminimality is at most one. The results have a natural interpretation in terms of invariant or periodic subvarieties of algebraic dynamical systems and $σ$-varieties. Surprisingly, the proofs of these results rely on the classification of finite simple groups. |
| title | When any four solutions are independent |
| topic | Logic Algebraic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2510.08387 |