Power Series Arithmetic with the BPAS Library

Multivariate Power Series in Maple

“…To solve these systems we proceed by solving them modulo successive powers of M, following the proof of Theorem 1 in [6]. Notice that solving modulo successive powers of M is precisely the same as computing homogeneous parts of increasing degree.…”

“…, X n ]][X n+1 ] is a UPoPS object. Following the ideas of [6], we compute the factors of f in a lazy fashion. Algorithm 2 proceeds through iterative applications of Taylor shift and Weierstrass Preparation Theorem in order to create one factor of f at a time.…”

“…Our implementation follows the lazy evaluation scheme of multivariate power series in the BPAS library [3]. The multivariate power series of BPAS, written in the C language, is discussed in [6] and extends upon the work of the PowerSeries subpackage of the RegularChains Maple library [2,13]. The PowerSeries package is the only preexisting implementation of multivariate power series integrated in Maple.…”

“…The PowerSeries package is the only preexisting implementation of multivariate power series integrated in Maple. In [6], it is shown that the BPAS implementation provides exceptional performance, surpassing that of the PowerSeries package, the basic Maple function mtaylor, and the multivariate power series available in SageMath [16] by multiple orders of magnitude.…”

“…In this section we review the basic properties of formal power series and univariate polynomials over those series, following G. Fischer in [8]. While various proofs of Theorems 1 of 2 can be found in the literature, the proofs given in [6] are constructive and support our implementation. Throughout this paper, N denotes the semi-ring of non-negative integers and K an algebraic number field.…”