Degree complexity of birational maps related to matrix inversion: symmetric case

Abstract

For q ≥ 3, we let S q denote the projectivization of the set of symmetric q × q matrices with coefficients in C. We let I (x) = (x i, j ) −1 denote the matrix inverse, and we let J (x) = (x −1 i, j ) be the matrix whose entries are the reciprocals of the entries of x. We let K |S q = I • J : S q → S q denote the restriction of the composition I • J to S q . This is a birational map whose properties have attracted some attention in statistical mechanics. In this paper we compute the degree complexity of K |S q , thus confirming a conjecture of Angles d'

Some interesting birational morphisms of smooth affine quadric 3-folds *

“…Besides being of interest in pure mathematics, birational maps appear naturally in some physical models (in lattice statistical mechanics), and their dynamical degrees are an indication of the complexity of these models, see e.g. [2,3,[8][9][10]61].…”