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Formal Laurent series in several variables.

Ainhoa Aparicio Monforte1, Manuel Kauers2

  • 1Université Lille 1 Sciences et Technologies, 59655 Villeneuve d'Ascq, Cedex, France.

Expositiones Mathematicae
|October 20, 2015
PubMed
Summary
This summary is machine-generated.

This study introduces fields of formal infinite series in multiple variables, extending the concept of formal Laurent series. It details operations like addition, multiplication, division, and composition, including an implicit function theorem.

Keywords:
Formal power series

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Area of Science:

  • Mathematics
  • Algebraic Geometry
  • Commutative Algebra

Background:

  • Formal Laurent series in one variable are a fundamental concept in mathematics.
  • Generalizing these series to multiple variables presents unique algebraic challenges.

Purpose of the Study:

  • To construct and define fields of formal infinite series in several variables.
  • To generalize the classical notion of formal Laurent series.

Main Methods:

  • Development of algebraic structures for formal infinite series in multiple variables.
  • Extension of field operations (addition, multiplication, division) to these series.
  • Investigation of series composition and an implicit function theorem.

Main Results:

  • Successful construction of fields of formal infinite series in several variables.
  • Demonstration of the applicability of standard field operations.
  • Formulation of an implicit function theorem for these generalized series.

Conclusions:

  • The study provides a rigorous framework for formal infinite series in multiple variables.
  • This generalization expands the toolkit for algebraic and analytic investigations.
  • The implicit function theorem offers new possibilities for solving equations involving these series.