On k-simple forms of first-order linear differential systems and their computation

In this paper, we develop a direct method for computing a k-simple form (see Pflügel, 2000) of a singular linear differential system of first-order. The k-simple forms give information on the integer slopes of the Newton polygon of the system and are useful in the construction of its formal solutions (see Barkatou and Pflügel, 1998; Pflügel, 2000). We study the arithmetic complexity of our algorithm which has been implemented in Maple and we illustrate it with some examples. Finally, we show how using this algorithm one can find the minimal Poincaré-rank and the formal invariants of the system.