Result: representation of algebraic functions as power and fractional power series of special type: Representation of algebraic functions as power and fractional power series of special type

Let \(K\) be a field and let \(f\in K[x,y]\). The equation \(f(x,y)= 0\) defines a function \(y(x)\) which expands to a fractional power series, or Puiseux series, \[ y(x)= \sum^\infty_{n= n_0} \alpha_n(x- x_0)^{n/\varepsilon},\tag{1} \] where \(\varepsilon\in \mathbb{N}\), \(n_0\in\mathbb{Z}\). The purpose of this paper is to describe a Maple procedure by means of which \(y(x)\) can be expanded into a ``nice'' expansion. Nice means that for some extension \(\widetilde K\) of the field \(K\), the sequence \(\{\alpha_n\}\) is polynomial, rational, hypergeometric, \(m\)-sparse, or \(m\)-sparse \(m\)-hypergeometric. This procedure can construct an expansion (1) at a given point \(x_0\), as well as determine points where a nice expansion is possible. Maple package Algcurves is used to compute \(\varepsilon\), \(n_0\), \(\alpha_0,\dots, \alpha_\nu\). This work follows \textit{D. V. Chudnovsky} and \textit{G. V. Chudnovsky's} method [J. Complexity 2, 271-294 (1986; Zbl 0629.68038); ibid. 3, 1-25 (1987; Zbl 0656.34003)], which allows to find a large number of coefficients in expansion (1) constructing a solution to a linear ordinary differential equation \(Ly(x)= 0\) (Maple Slode package). To construct this differential equation the authors implement an algorithm in the form of Maple procedure. Five examples in Maple 6 are presented for functions expandable into nice series.