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

摘要

For q ≥ 3, we let S _q denote the projectivization of the set of symmetric q × q matrices with coefficients in ?. We let I(x) = (x _(i,j)) 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{pipe}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{pipe}S _q thus confirming a conjecture of Angles d'Auriac et al. (J Phys A Math Gen 39:3641-3654, 2006).