The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree

Matthew England, James H. Davenport

Research output: Chapter in Book/Report/Conference proceedingConference proceeding

11 Citations (Scopus)
5 Downloads (Pure)

Abstract

Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be improved by adapting to take advantage of any equational constraints (ECs): equations logically implied by the input. Intuitively, we expect the double exponent in the complexity to decrease by one for each EC. In ISSAC 2015 the present authors proved this for the factor in the complexity bound dependent on the number of polynomials in the input. However, the other term, that dependent on the degree of the input polynomials, remained unchanged. In the present paper the authors investigate how CAD in the presence of ECs could be further refined using the technology of Gr¨obner Bases to move towards the intuitive bound for polynomial degree.
Original languageEnglish
Title of host publicationComputer Algebra in Scientific Computing
EditorsVladimir P. Gerdt, Wolfram Koepf, Werner M. Seiler, Evgenii V. Vorozhtsov
Place of PublicationSwitzerland
PublisherSpringer Verlag
Pages172-192
Number of pages21
ISBN (Electronic)978-3-319-45641-6
ISBN (Print)978-3-319-45640-9
DOIs
Publication statusPublished - 9 Sep 2016
EventInternational Workshop on Computer Algebra in Scientific Computing - Bucharest, Romania
Duration: 19 Sep 201623 Sep 2016

Publication series

NameLecture Notes in Computer Science
PublisherSpringer Verlag
Volume9890
ISSN (Print)0302-9743

Conference

ConferenceInternational Workshop on Computer Algebra in Scientific Computing
CountryRomania
CityBucharest
Period19/09/1623/09/16

Bibliographical note

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-45641-6_12

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Lecture Notes in Computer Science ISSN 0302-9743

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    England, M., & Davenport, J. H. (2016). The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree. In V. P. Gerdt, W. Koepf, W. M. Seiler, & E. V. Vorozhtsov (Eds.), Computer Algebra in Scientific Computing (pp. 172-192). (Lecture Notes in Computer Science ; Vol. 9890). Switzerland: Springer Verlag. https://doi.org/10.1007/978-3-319-45641-6_12