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 nonzero terms of q with total degree d. Second, a non-negative integer, called the precision of q, which ensures that all nonzero 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 nonzero 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, or 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 command PuiseuxSeries creates a Puiseux 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, and Variables access data from a power series or a univariate polynomial over power series. The command Truncate can also be used on a Puiseux series.

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

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

The command IsUnit determines if a power series is invertible.

The commands ApproximatelyEqual and ApproximatelyZero determine equality and equality with zero up to some precision.

The command Substitute substitutes values into a power series.

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

The command PuiseuxFactorize factorizes a univariate polynomial over power series with Puiseux series coefficients.

The command TschirnhausenTransformation applies a Tschirnhausen transfomration to a univariate polynomial over power series with Puiseux coefficients.

The TaylorShift command performs a Taylor shift of a univariate polynomial over power series.

The command GetAnalyticExpression returns the analytic expression of a power series, a univariate polynomial over power series, or a Puiseux series, if known.

The commands ChangeOfVariables, GetMonomial, GetOrder, GetPowerSeries, GetPowerSeriesOrder, GetPuiseuxSeriesOrder, and GetRays retrieve information on a Puiseux series.

The command SetPuiseuxBound sets a bound that is used in computing the Puiseux factorization.

The commands SetNonzeroPowerSeriesDegreeBound and SetSmallestTermDegreeBound set bounds that are used in computing the inverse of a Puiseux series.

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

The following is a list of available commands.

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. Vol. 12291 (2020): 108-128.

Mohammadali Asadi, Alexander Brandt, Mahsa Kazemi, Marc Moreno Maza, and Erik Postma. "Multivariate Power Series in Maple." Corless R.M., Gerhard J., Kotsireas I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science (CCIS), Vol. 1414 Springer (2021): 48-66.

Parisa Alvandi, Massoud Ataei, Mahsa Kazemi, Marc Moreno Maza. "On the extended Hensel construction and its application to the computation of real limit points J. of Symbolic Computation, Vol. 98 (2020): 120-162.

Marc Moreno Maza, Erik Postma. "Substituting Units into Multivariate Power Series." Proc. of MC 2021. 2021.

K. J. Nowak. "Some elementary proofs of Puiseux's theorems." Univ. Iagel. Acta Math. Vol. 38 (2000): 279-282.

Alexander Brandt, Marc Moreno Maza. "On the Complexity and Parallel Implementation of Hensel’s Lemma and Weierstrass Preparation." Proc. of CASC 2021. Vol. 12865. Lecture Notes in Computer Science. Springer, 2021, pp. 78–99.

M. Calder, J. P. González Trochez, M. Moreno Maza, and E. Postma. "Algorithms for multivariate Laurent series." Chapter 4 in the MSc Thesis of Juan Pablo González Trochez, submitted to MC 2022. 2022.

A. A. Monforte and M. Kauers. "Formal Laurent series in several variables." Expositiones Mathematicae. Vol. 31 No. 4 (2013): 350–367.

The MultivariatePowerSeries package was introduced in Maple 2021.

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