Arnold normal form and formal reduction of system of linear difference equations
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.
scientific article; zbMATH DE number 1098631
scientific article