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arXiv · 2306.11108

Invariant rational functions under rational transformations

Jason Bell·Rahim Moosa·Matthew Satriano
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Abstract

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.

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Jason Bell, Rahim Moosa, Matthew Satriano. 2024-03-11. Invariant rational functions under rational transformations. https://arxiv.org/abs/2306.11108

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