Applicable Algebra in Engineering, Communication.pdf

AAECC 7, 21-26 (1996) MECC Applicable Algebra in Engineering, Communication and Computing 9 Springer-Verlag 1996 Closed Form Laurent-Puiseux Series of Algebraic Functions Wolfram Koepf Konrad-Zuse-Zentrum ffir Informationstechnik Berlin, Heilbronner Str. 10, D-10711 Berlin, Germany Received December 9, 1993; revised version January 6, 1995 Abstract. There are several well-known algorithms to calculate the Puiseux series developments of the branches of an algebraic function. None of them, however, generates the series in closed form, even in those cases where such a formal result is available. They produce, instead, truncated series, and give information that can be used to handle the series as streams. Here we give a solution to the given problem. We combine an algorithm ofD. V. and G. V. Chudnovsky that transforms the given algebraic equation into a differential equation for the function, and further into a recurrence equation for the Puiseux coefficients, with an algorithm of Koepf which in the case of hypergemetric type results in the formal series. A finite linear recurrence equation is optimal for a representation by streams. D. V. and G.V. Chudnovsky point out that their algorithm requires only O(M) field operations if M is the order of the number of series terms considered. However, from a practical point of view, it is of importance that the complexity of the resulting recurrence equa t ion - as well as of the differential equa t ion - can be extremely high, a fact, which we illustrate by an example. It turns out, that many alge- braic functions of low order with a sparse representation are of hypergeometric type, and so closed form representations for the corresponding series can be given. Keywords: formal power series, Laurent-Puiseux series, closed forms, hypergeometric terms and functions, functions of hypergeometric type, holonomic linear differential and recurrence equations. 1 Introduction We consider algebraic fun