On the complexity of some birational transformations

TL;DR

This paper investigates the complexity of certain birational transformations derived from matrix inversions using algebraic, singularity, and numerical methods, leading to a conjecture about their complexity.

Contribution

It introduces three distinct approaches to analyze the complexity of birational maps from matrix inversions and proposes a conjecture based on corroborated results.

Findings

Different methods yield consistent results

A conjecture on the complexity of matrix-inversion-based maps

Insights into the algebraic and singularity structures of these maps

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.