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

Abstract

For q ≥ 3, we let \({\mathcal{S}_q}\) denote the projectivization of the set of symmetric q × q matrices with coefficients in \({\mathbb{C}}\). We let \({I(x)=(x_{i,j})^{-1}}\) denote the matrix inverse, and we let \({J(x)=(x_{i,j}^{-1})}\) be the matrix whose entries are the reciprocals of the entries of x. We let \({K|\mathcal{S}_q=I\circ J:~\mathcal{S}_q\rightarrow \mathcal{S}_q}\) denote the restriction of the composition I ◦ 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 et al. (J Phys A Math Gen 39:3641–3654, 2006).