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Degree complexity of birational maps related to matrix inversion. (English) Zbl 1205.15009

For a \(q \times q\) matrix \(x = (x _{i, j })\), let \({J(x)=(x_{i,j}^{-1})}\) be the Hadamard inverse, which takes the reciprocal of the elements of \(x\). Assume that \({I(x)=(x_{i,j})^{-1}}\) denotes the matrix inverse, and define \({K=I\circ J}\) to be the birational map obtained from the composition of these two involutions. The authors consider the iterates \({K^n=K\circ\cdots\circ K}\) and determine the degree complexity of \(K\), which is the exponential rate of degree growth \({\delta(K)=\lim_{n\to\infty}\left( deg(K^n) \right)^{1/n}}\) of the degrees of the iterates. Earlier studies of this map were restricted to cyclic matrices, in which case \(K\) may be represented by a simpler map. The authors show that for general matrices the value of \(\delta (K)\) is equal to the value conjectured by J.-Ch. Anglès d’Auriac, J.-M. Maillard and C. M. Viallet [J. Phys. A, Math. Gen. 39, No. 14, 3641–3654 (2006; Zbl 1086.14504)].

MSC:

15A09 Theory of matrix inversion and generalized inverses
14E99 Birational geometry

Citations:

Zbl 1086.14504
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References:

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