The Multivariate Power Series Package in Maple 2024.

While symbolic computation is the realm of exact methods, this field was able to develop solutions to approximate problems, which can be used to provide approximate answers to problems that are either intractable or too expensive to solve exactly. A well-known example is the so-called symbolic Newton iteration method for approximating the solutions of algebraic equations. At the heart of these methods is the manipulation of formal multivariate power series.The MultivariatePowerSeries library in Maple provides formal, Laurent and Puiseux series in several variables. The implementation of those series is based on the paradigm of lazy evaluation (or call-by-need). Not only arithmetic operations (addition, multiplication, inversion) are offered, but univariate polynomial over multivariate series are provided by this library. The user can factorize such polynomials in a number of ways, using either Weierstrass Preparation Theorem, Hensel Lemma, Puiseux Theorem and the Extended Hensel Construction. In this talk, we will give a tour of these facilities and illustrate their usage with a number of applications.