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

Abstract

For q ≥ 3, we let Sq denote the projectivization of the set of symmetric q × q matrices with coefficients in C. We let I(x)=(xi,j)−1 denote the matrix inverse, and we let J(x)=(x−1i,j) be the matrix whose entries are the reciprocals of the entries of x. We let K|Sq=I∘J: Sq→Sq denote the restriction of the composition I ◦ J to Sq. This is a birational map whose properties have attracted some attention in statistical mechanics. In this paper we compute the degree complexity of K|Sq, thus confirming a conjecture of Angles d’Auriac et al. (J Phys A Math Gen 39:3641–3654, 2006).