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A geometric approach to cylindrical algebraic decomposition.

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Title: A geometric approach to cylindrical algebraic decomposition.
Authors: Chen, Rizeng1 (AUTHOR)
Source: Mathematics of Computation. Jul2026, Vol. 95 Issue 360, p2025-2059. 35p.
Subjects: Geometric approach, Mathematic morphism, Algorithms, Semialgebraic sets, Algebraic geometry, Semianalytic sets
Abstract: Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this paper, we revisit this problem from a more geometric perspective, where the construction of cylindrical algebraic decomposition is related to the study of morphisms between real varieties. It is showed that the geometric fiber cardinality (geometric property) decides the existence of semi-algebraic continuous sections (semi-algebraic property). As a result, all equations can be systematically exploited in the projection phase, leading to a new simple algorithm whose efficiency is demonstrated by experimental results. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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  Availability: 0
Header DbId: egs
DbLabel: Engineering Source
An: 192903043
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
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  Data: A geometric approach to cylindrical algebraic decomposition.
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+of+Computation%22">Mathematics of Computation</searchLink>. Jul2026, Vol. 95 Issue 360, p2025-2059. 35p.
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  Data: Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this paper, we revisit this problem from a more geometric perspective, where the construction of cylindrical algebraic decomposition is related to the study of morphisms between real varieties. It is showed that the geometric fiber cardinality (geometric property) decides the existence of semi-algebraic continuous sections (semi-algebraic property). As a result, all equations can be systematically exploited in the projection phase, leading to a new simple algorithm whose efficiency is demonstrated by experimental results. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1090/mcom/4099
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 35
        StartPage: 2025
    Subjects:
      – SubjectFull: Geometric approach
        Type: general
      – SubjectFull: Mathematic morphism
        Type: general
      – SubjectFull: Algorithms
        Type: general
      – SubjectFull: Semialgebraic sets
        Type: general
      – SubjectFull: Algebraic geometry
        Type: general
      – SubjectFull: Semianalytic sets
        Type: general
    Titles:
      – TitleFull: A geometric approach to cylindrical algebraic decomposition.
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            NameFull: Chen, Rizeng
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          Dates:
            – D: 01
              M: 07
              Text: Jul2026
              Type: published
              Y: 2026
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              Value: 95
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              Value: 360
          Titles:
            – TitleFull: Mathematics of Computation
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