Arnold normal form and formal reduction of system of linear difference equations (Q1376588)

The author presents an algorithm to obtain Turrittin's formal reduction of linear difference equations of the forms \(Y(x+1)=x^cA(x)Y(x)\) or \(Y(x+1)-Y(x)=x^{-r/s}A(x)Y(x)\), where \(c\) is a rational number, \(r\) and \(s\) are two positive integers, and the \(n\times n\) matrix \(A(x)\) admits the formal expansion \(A(x)=A_0+A_1 x^{-1/s}+A_2 x^{-2/s}+\ldots\), with \(A_0\neq0\) [cf. \textit{H. L. Turrittin}, Bol. Soc. Mat. Mex., II. Ser. 5, 255-264 (1960; Zbl 0100.08201)]. In particular, \(\det{A(x)}\neq0\) in the first system. In the nilpotent case, the Arnold normal form of matrices depending on parameters is used [cf. \textit{V. I. Arnold}, Usp. Mat. Nauk 26, No. 2(158), 101-114 (1971; Zbl 0258.15011)]. It is shown that for systems in the second form, the exponential part in a formal fundamental matrix solution \(Y(x)=F(x^{-1/s'})x^C\exp(Q(x^{1/s'})\) depends only on the matrix coefficients \(A_0,\ldots,A_{m(s-r)-1}\). Similarly, for systems in the first form, the \(\Gamma\) part of the formal fundamental matrix solution \(Y(x)=F(x^{-1/s'})x^C\exp(Q(x^{1/s'})R^x\Gamma(x)^M\) depends only on \(A_0,\ldots,A_m\), and, generally, the irregular part depends only on \(A_0,\ldots,A_{m+ns-1}\). These results are important from an algorithmic point of view and the bounds are best possible.