Submission date: 19 June 2023

Abstract:

Let X be a complex algebraic variety equipped with a birational map φ: X → X. A new quantity measuring the interaction of (X,φ) with trivial dynamical systems is introduced; the eventual 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 is bounded above by the maximum dimension of a translation variety appearing as a dominant equivariant rational image. Moreover, if (X,φ) has no nonconstant invariant rational functions then these two quantities agree. 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.

Mathematics Subject Classification: 14E07, 12H10, 12L12

Keywords and phrases: Invariant rational functions and translation varieties.

Full text arXiv 2306.11108: pdf, ps.