On invariant rational functions under rational transformations

• Jason Bell ^[1] ; Rahim Moosa ^[1] ; Matthew Satriano ^[1]
  1. [1] University of Waterloo

     University of Waterloo

     Canadá

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 3, 2024, págs. 1-23
• Idioma: inglés
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• Resumen

  • Let X be an algebraic variety equipped with a dominant rational map φ : X 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 (X × Y , φ × ψ), as ψ : Y Y ranges over all dominant rational maps on algebraic varieties. It is shown that this birational invariant agrees with the maximum dim X where (X, φ) (X , φ )is a dominant rational equivariant map andφ 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.