On the complexity of some birational transformations

Abstract

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis, and the third method is more numerical, using integer arithmetics. Each method has its own domain of application, but they give corroborating results, and lead us to a conjecture on the complexity of a class of maps constructed from matrix inversions.

References in corpus (2)

• A classification of four-state spin edge Potts models
• Exact solution for a matrix dynamical system with usual and Hadamard inverses

Cited by in corpus (13)

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• Height growth of solutions and a discrete Painlevé equation
• Spectral interpretations of dynamical degrees and applications
• Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices
• Degree complexity of birational maps related to matrix inversion: Symmetric case
• Zero Entropy for Some Birational Maps of C^2
• Dynamical Classification of a Family of Birational Maps of C^2 via Algebraic Entropy
• A central limit theorem for the degree of a random product of Cremona transformations
• Degree Growth of Rational Maps Induced from Algebraic Structures
• Degree Complexity of Matrix Inversion