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

2010/05/25 by Tuyen Trung Truong, Truong, Tuyen Trung
Mathematics · #32H50 #37F99 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:32H50 #msc:37F99

paper · pdf · doi:10.48550/arxiv.1005.4520

14 pages

arxiv created 2010/11/04 · arxiv updated 2010/11/05

Abstract

For q≥ 3, we let S_q denote the projectivization of the set of symmetric q× q matrices with coefficients in 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|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|S_q, thus confirming a conjecture of Angles d'Auriac, Maillard, and Viallet in [J. Phys. A: Math. Gen. 39 (2006), 3641--3654].

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• Degree Complexity of Matrix Inversion
• Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices
• Degree complexity of a family of birational maps: II. Exceptional cases
• Polynomial Complexity of Inversion of sequences and Local Inversion of Maps
• Equations and multidegrees for inverse symmetric matrix pairs