For \(n\in\mathbb N\) the ring of formal power series in \(n\) commuting variables with complex coefficients is denoted by \(\mathfrak F_n\). The maximal ideal of this ring is denoted by \(\mathfrak M_n\) and consists of those power series with zero constant term. Let \(\mathfrak C_n=(\mathfrak M_n)^n\) and \(\mathfrak G_n\) be the group of formally invertible elements in \(\mathfrak C_n\). Alternatively, \(\mathfrak G_n\) can be viewed as the group of \(\mathbb C\)-algebra automorphisms of \(\mathfrak F_n\).
In this article the authors continue their investigation into the factorisation of elements in \(\mathfrak G_n\) into products of reversibles or involutions. An element is defined to be an involution if it is equal to its inverse and reversible if it is conjugate to its inverse. In analysing which elements can be so written the authors consider the linear part \(L(F)\) of an element \(F\in\mathfrak G_n\) which is defined via the natural homomorphism from \(\mathfrak G_n\) to \(\text{GL}(n,\mathbb C)\).
For \(n\geq 2\) and \(F\in\mathfrak G_n\) the authors prove that the following statements are equivalent. (1) \(F\) is a product of reversibles. (2) \(L(F)\) has determinant \(\pm 1\). (3) \(F\) is the product of \(2+3\lceil\log_2n\rceil\) reversibles. (4) \(F\) is the product of \(9+6\lceil\log_2n\rceil\) involutions.
The first author had previously considered the case when \(n=1\) where, in contrast, there exist reversible elements that cannot be expressed as a finite number of involutions [A. G. O’Farrell, Comput. Methods Funct. Theory 8, No. 1, 173-193 (2008; Zbl 1232.20045)].
The authors pose a number of questions, including whether the bounds given above are sharp.