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

摘要

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 ${K|mathcal{S}_q=Icirc J:~mathcal{S}_qrightarrow mathcal{S}_q}$ 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).