On invariant rational functions under rational transformations
type: Item
instance of: scholarly article
title: On invariant rational functions under rational transformations
author: Jason P. Bell
author: Rahim Moosa
author: Matthew Satriano
published in: Selecta Mathematica. New Series
publication date: 2024-06-14T00:00:00Z
review text: The authors consider rational dynamical systems \((X,\phi)\) over an algebraically closed field \(k\), where \(X\) is an irreducible quasi-projective variety over \(k\) and \(\phi:X\dashrightarrow X\) is a dominant rational map over \(k\). The main result of the paper is that two dimensions defined by the authors coincide. The first one is the maximum number of new algebraically independent invariant rational functions on \((X\times Y, \phi\times\psi)\), as \(\psi : Y \dashrightarrow Y\) ranges over all dominant rational maps on algebraic varieties. It is called stabilised algebraic dimension of \((X, \phi)\) and denoted by \(\mathrm{sadim}(X, \phi)\). The second dimension is the maximum \(n\geq 0\) such that there is a translational dynamical system \((Y,\psi)\) over \(k\) of dimension \(n\), and a dominant rational equivariant map \((X, \phi) \dashrightarrow (Y,\psi)\). It is called the translation dimension of \((X, \phi)\) and denoted by tdim\((X, \phi)\).
reviewed by: Ivo M. Michailov
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 12H10
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 7865504
zbMATH Keywords: algebraic dynamics
zbMATH Keywords: rational transformations
zbMATH Keywords: difference fields
MaRDI profile type: Publication
cites work: Remarks on endomorphisms and rational points
cites work: Meromorphic fibrations on certain manifolds with trivial canonical bundle
cites work: Q5563439
cites work: The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings.
cites work: On a dynamical version of a theorem of Rosenlicht
cites work: Topologies and structures of the Cremona groups
cites work: Invariant hypersurfaces in holomorphic dynamics
cites work: Morphisms between Cremona groups and characterization of rational varieties
cites work: Difference fields and descent in algebraic dynamics. I
cites work: Difference fields and descent in algebraic dynamics. II
cites work: Model theory of difference fields
cites work: Sous-groupes algébriques de rang maximum du groupe de Cremona
cites work: Definable groups of partial automorphisms
cites work: Invariant varieties for polynomial dynamical systems
cites work: Some Basic Theorems on Algebraic Groups
cites work: On Algebraic Groups of Transformations
cites work: Q5434295
DOI: 10.1007/S00029-024-00940-8
description: scientific article; zbMATH DE number 7865504