Invariant rational functions under rational transformations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910363315863552 |
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| author | Bell, Jason Moosa, Rahim Satriano, Matthew |
| author_facet | Bell, Jason Moosa, Rahim Satriano, Matthew |
| contents | Let $X$ be an algebraic variety equipped with a dominant rational self-map $ϕ:X\to X$. A new quantity measuring the interaction of $(X,ϕ)$ with trivial dynamical systems is introduced; the stabilised algebraic dimension of $(X,ϕ)$ captures the maximum number of new algebraically independent invariant rational functions on the cartesian product of $(X, ϕ)$ and $(Y, ψ)$, as $(Y,ψ)$ ranges over all algebraic dynamical systems. It is shown that this birational invariant agrees with the maximum dimension of a dominant equivariant rational image $(X',ϕ')$ where $ϕ'$ is part of an algebraic group action on $X'$. As a consequence, it is deduced that if some cartesian power of $(X,ϕ)$ admits a nonconstant invariant rational function, then already the second cartesian power does. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11108 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invariant rational functions under rational transformations Bell, Jason Moosa, Rahim Satriano, Matthew Algebraic Geometry Dynamical Systems Logic 14E07, 12H10, 12L12 Let $X$ be an algebraic variety equipped with a dominant rational self-map $ϕ:X\to X$. A new quantity measuring the interaction of $(X,ϕ)$ with trivial dynamical systems is introduced; the stabilised algebraic dimension of $(X,ϕ)$ captures the maximum number of new algebraically independent invariant rational functions on the cartesian product of $(X, ϕ)$ and $(Y, ψ)$, as $(Y,ψ)$ ranges over all algebraic dynamical systems. It is shown that this birational invariant agrees with the maximum dimension of a dominant equivariant rational image $(X',ϕ')$ where $ϕ'$ is part of an algebraic group action on $X'$. As a consequence, it is deduced that if some cartesian power of $(X,ϕ)$ admits a nonconstant invariant rational function, then already the second cartesian power does. |
| title | Invariant rational functions under rational transformations |
| topic | Algebraic Geometry Dynamical Systems Logic 14E07, 12H10, 12L12 |
| url | https://arxiv.org/abs/2306.11108 |