Doing Algebraic Geometry with the RegularChains Library

  • Parisa Alvandi
  • Changbo Chen
  • Steffen Marcus
  • Marc Moreno Maza
  • Éric Schost
  • Paul Vrbik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8592)

Abstract

Traditionally, Groebner bases and cylindrical algebraic decomposition are the fundamental tools of computational algebraic geometry. Recent progress in the theory of regular chains has exhibited efficient algorithms for doing local analysis on algebraic varieties. In this note, we present the implementation of these new ideas within the module AlgebraicGeometryTools of the RegularChains library. The functionalities of this new module include the computation of the (non-trivial) limit points of the quasi-component of a regular chain. This type of calculation has several applications like computing the Zarisky closure of a constructible set as well as computing tangent cones of space curves, thus providing an alternative to the standard approaches based on Groebner bases and standard bases, respectively. From there, we have derived an algorithm which, under genericity assumptions, computes the intersection multiplicity of a zero-dimensional variety at any of its points. This algorithm relies only on the manipulations of regular chains.

Keywords

Algebraic geometry regular chains local analysis 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Parisa Alvandi
    • 1
  • Changbo Chen
    • 2
  • Steffen Marcus
    • 3
  • Marc Moreno Maza
    • 1
  • Éric Schost
    • 1
  • Paul Vrbik
    • 1
  1. 1.University of Western OntarioCanada
  2. 2.Chongqing Institute of Green and Intelligent TechnologyCASChina
  3. 3.The College of New JerseyEwingUSA

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