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Invariant rational functions under rational transformations

Jason Bell, Rahim Moosa, Matthew Satriano

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.

Invariant rational functions under rational transformations

Abstract

Let be an algebraic variety equipped with a dominant rational self-map . A new quantity measuring the interaction of with trivial dynamical systems is introduced; the stabilised algebraic dimension of captures the maximum number of new algebraically independent invariant rational functions on the cartesian product of and , as ranges over all algebraic dynamical systems. It is shown that this birational invariant agrees with the maximum dimension of a dominant equivariant rational image where is part of an algebraic group action on . As a consequence, it is deduced that if some cartesian power of admits a nonconstant invariant rational function, then already the second cartesian power does.
Paper Structure (8 sections, 22 theorems, 42 equations)

This paper contains 8 sections, 22 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Algebraic dimension
  3. Algebraic reduction
  4. Families of invariant subvarieties
  5. Translations
  6. Stabilised algebraic dimension for translations
  7. Families of rational transformations
  8. The proof

Key Result

Theorem 1

Suppose $(X,\phi)$ is a rational dynamical system over an algebraically closed field $k$. Then $\operatorname{sadim}(X,\phi)=\operatorname{tdim}(X,\phi)$.

Theorems & Definitions (52)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem
  • Corollary A
  • Corollary B
  • Remark 1.6
  • Lemma 2.1
  • ...and 42 more