Multivariate Power Series in Maple

Asadi, Mohammadali; Brandt, Alexander; Kazemi, Mahsa; Maza, Marc Moreno; Postma, Erik

doi:10.1007/978-3-030-81698-8_4

Cited by 4 publications

(12 citation statements)

References 11 publications

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“…. , 𝑞r 𝑚 }) multiplied by 1 𝑞 . As we shall see in Section 3, this plays an important role in the definition of our Puiseux series object, that is, in the data-structure representing our Puiseux series.…”

Section: Conesmentioning

confidence: 99%

“…In this document, following the ideas exposed by Monforte and Kauers in [4], we report on a first implementation of multivariate Laurent series and multivariate Puiseux series in Maple. This implementation takes advantage of the already existing MultivariatePowerSeries package presented in [2].To implement multivariate Laurent series, we first need to select an additive total order, such that we can compare exponent vectors. We choose the well-known graded reverse lexicographic order, or grevlex order, for short.…”

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confidence: 99%

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Laurent Series and Puiseux Series in Maple

Gonzalez Trochez,

Moreno Maza,

Postma

et al. 2023

Maple Trans.

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Let K be an algebraically closed field of characteristic zero. The field of fractions of the ring of formal multivariate power series over K, is called the field of formal multivariate Laurent series. In this document, we follow the ideas introduced by Monforte and Kauers in their paper Formal Laurent Series in Several Variables. Our objective is to report on a first implementation of formal multivariate Laurent series inside of Maple, and explain the challenges we had to overcome. In order to accomplish this goal, we make use of the already existing MultitivariatePowerSeries package, and its lazy evaluation scheme. In particular, we expose our ideas for adding and multiplying Laurent series with support inside different cones, where the support of a Laurent series is the set of all exponents of all non-zero monomials of our series. We also describe our biggest challenge, how to invert a Laurent series. Unfortunately, this problem cannot be completely solved in a lazy evaluation context. We describe some situations where we can solve the problem completely; our approach for the cases that fall outside of these situations; and how we let the user customize this approach, trading off between speed and the likelihood of an incorrect result. The algebraic closure of the field of formal multivariate Laurent series is call the field of formal multivariate Puiseux series. As an extension of our current work, we also present our ideas for an implementation of a multivariate Puiseux series object inside of Maple.

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Section: Conesmentioning

confidence: 99%

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Laurent Series and Puiseux Series in Maple

Gonzalez Trochez,

Moreno Maza,

Postma

et al. 2023

Maple Trans.

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“…We will often write simply 0 for a vector of zeroes; so we might write the previous equation as 𝑎 (0) = 𝑎 0 . This is the setting in which we deal with power series in the MultivariatePowerSeries package [3], introduced in Maple 2021 [12]. The package deals with formal power series over the complex numbers.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…Algebraic power series are formal power series which satisfy a univariate polynomial equation over a ring of multivariate polynomials. They occur for instance in the application of the Weierstrass Preparation Theorem and can be manipulated in an algorithmic fashion, in this context as shown in [3,5]. See also [1] for a more general treatment of algebraic power series.…”

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confidence: 99%