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

Abstract

For $q\ge 3$, we let ${\mathcal{S}}_q$ denote the projectivization of the set of symmetric $q\times q$ matrices with coefficients in $\mathbb{C}$. We let $I(x)=(x_{i,j}{)}^{-1}$ denote the matrix inversion, and we let $J(x)=(x_{i,j}^{-1})$ be the matrix whose entries are the reciprocals of the entries of $x$. We let $K|{\mathcal{S}}_q=I\circ J:{\mathcal{S}}_q\to {\mathcal{S}}_q$ denote the restriction of the composition $I\circ J$ to ${\mathcal{S}}_q$. This is a birational map whose properties have attracted some attention in statistical mechanics. In this paper we compute the degree complexity of $K|{\mathcal{S}}_q$, thus confirming a conjecture of Angles d'Auriac, Maillard, and Viallet in [J. Phys. A: Math. Gen. 39 (2006), 3641--3654].

14 pages