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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2510.16314 |
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| _version_ | 1866917024258588672 |
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| author | Kamensky, Moshe Moosa, Rahim |
| author_facet | Kamensky, Moshe Moosa, Rahim |
| contents | Given an algebraic difference equation of the form \[σ^n(y)=f\big(y, σ(y),\dots,σ^{n-1}(y)\big)\] where $f$ is a rational function over a field $k$ of characteristic zero on which $σ$ acts trivially, it is shown that if there is a nontrivial algebraic relation amongst any number of $σ$-disjoint solutions, along with their $σ$-transforms, then there is already such a relation between three solutions. Here ``$σ$-disjoint" means $a\neqσ^r(b)$ for any integer $r$. A weaker version of the theorem, where ``three" is replaced by $n+4$, is also obtained when $σ$ acts non-trivially on $k$. Along the way a number of other structural results about primitive rational dynamical systems are established. These theorems are deduced as applications of a detailed model-theoretic study of finite-rank quantifier-free types in the theory of existentially closed difference fields of characteristic zero. In particular, it is also shown that the degree of non-minimality of such types over fixed-field parameters is bounded by $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16314 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A transformal transcendence result for algebraic difference equations Kamensky, Moshe Moosa, Rahim Logic Algebraic Geometry Dynamical Systems 03C98 (Primary) 14E07, 37P05, 37P55, 12H10, 12L12 (Secondary) Given an algebraic difference equation of the form \[σ^n(y)=f\big(y, σ(y),\dots,σ^{n-1}(y)\big)\] where $f$ is a rational function over a field $k$ of characteristic zero on which $σ$ acts trivially, it is shown that if there is a nontrivial algebraic relation amongst any number of $σ$-disjoint solutions, along with their $σ$-transforms, then there is already such a relation between three solutions. Here ``$σ$-disjoint" means $a\neqσ^r(b)$ for any integer $r$. A weaker version of the theorem, where ``three" is replaced by $n+4$, is also obtained when $σ$ acts non-trivially on $k$. Along the way a number of other structural results about primitive rational dynamical systems are established. These theorems are deduced as applications of a detailed model-theoretic study of finite-rank quantifier-free types in the theory of existentially closed difference fields of characteristic zero. In particular, it is also shown that the degree of non-minimality of such types over fixed-field parameters is bounded by $2$. |
| title | A transformal transcendence result for algebraic difference equations |
| topic | Logic Algebraic Geometry Dynamical Systems 03C98 (Primary) 14E07, 37P05, 37P55, 12H10, 12L12 (Secondary) |
| url | https://arxiv.org/abs/2510.16314 |