The MultivariatePowerSeries package is a collection of commands for manipulating multivariate power series and univariate polynomials over multivariate power series.

The main algebraic functionalities of this package deal with arithmetic operations (addition, multiplication, inversion, evaluation), for both multivariate power series and univariate polynomials over multivariate power series, as well as factorization of such polynomials.

Every power series q is encoded by an object storing the following information. First, a procedure, called the power series generator, which, given a non-negative integer d returns all non-zero terms of q with total degree d. Second, a non-negative integer, called the precision of q, which ensures that all non-zero terms of q of degree less or equal to that precision have been computed and are stored. Third, an Array, called the data array of q such that, if all non-zero terms of q of degree i have been computed then they are stored at position i of that array, for all non-negative integers i.

The implementation of every arithmetic operation, such as addition, multiplication, inversion builds the resulting power series (sum, product or inverse) by creating its generator from the generators of the operands, which are called ancestors of the resulting power series. The coefficients of that resulting power series are computed only when truly needed. Once computed, they are stored in the data array of that power series, where they can be retrieved next time needed. When more terms (than those already stored in the data array) are needed, then the generator is invoked which, in turn, may invoke the generators of the ancestors.

The implementation of the factorization commands WeierstrassPreparation and HenselFactorize is also based on lazy evaluation (also known as calls-by-need). Each factorization command returns the factors as soon as enough information is discovered for initializing the data structures of the factors. The precision of each returned factor, that is, the common precision of its coefficients (which are power series) is zero. However the generator of each coefficient is known and, thus, the computation of more coefficients can be resumed when a higher precision is requested.

The commands PowerSeries and UnivariatePolynomialOverPowerSeries create power series and univariate polynomials over multivariate power series from objects like polynomials and sequences (given as functions). The commands GeometricSeries and SumOfAllMonomials create examples of power series.

The commands Display, SetDefaultDisplayStyle and SetDisplayStyle control the output format of multivariate power series and univariate polynomials over multivariate power series.

The commands HomogeneousPart, Truncate, GetCoefficient, Precision, Degree, MainVariable access data from a power series or a univariate polynomial over power series.

The commands UpdatePrecision and Copy manipulate data of multivariate power series and univariate polynomials over multivariate power series.

The commands Add, Negate, Multiply, Exponentiate, Inverse, Divide, EvaluateAtOrigin perform arithmetic operations on multivariate power series and univariate polynomials over multivariate power series. The functionality of the first six commands can also be accessed using the standard arithmetic operators when the arguments are power series.

The commands WeierstrassPreparation and HenselFactorize factorize univariate polynomials over multivariate power series.

When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series and univariate polynomials over power series. If you do, you may see invalid results.

Alexander Brandt, Mahsa Kazemi, Marc Moreno Maza "Power Series Arithmetic with the BPAS Library." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 12291, (2020): 108-128.

Mohammadali Asadi, Alexander Brandt, Mahsa Kazemi, Marc Moreno Maza, and Erik Postma: " Multivariate Power Series in Maple." Maple Conference 2020, Waterloo, Ontario, Canada, November 2-6, 2020, Communications in Computer and Information Science (CCIS) series - Springer 2020 (submitted).

The MultivariatePowerSeries package was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.