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Products of involutory matrices over rings. (English) Zbl 0836.15006

An involution in a group \(G\) is an element \(a\) of order 2. Let \(a \in \mathrm{GL}_n A\) be a matrix over a commutative ring \(A\) with 1 such that \((\det a)^2 = 1\). If \(a\) is cyclic, it can be written as a product of at most three involutions. The paper proves that when \(A\) satisfies the first Bass stable range condition, then \(a\) can be written as a product of at most five involutions. If in addition either \(n \leq 3\) or \(n = 4\) and \(\det a = - 1\), then \(a\) can be written as a product of at most four involutions. When \(A\) is a Dedekind ring of arithmetic type, the number of involutions needed to express \(a\) is uniformly bounded for any \(n \geq 3\). When \(A = \mathbb C[x]\) is the polynomial ring with complex coefficients, the number of involutions is unbounded for any \(n\geq 2\).

MSC:

15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A23 Factorization of matrices
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References:

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