Need Polynomial Systems Be Doubly-Exponential?

James H. Davenport, Matthew England

Research output: Chapter in Book/Report/Conference proceedingConference proceedingpeer-review

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Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables (see the classic papers of Mayr & Mayer from 1982 and Davenport & Heintz from 1988). Nevertheless, the Bezout bound tells us that number of zeros of a zero-dimensional system is singly-exponential in the number of variables. How should this contradiction be reconciled? We first note that Mayr and Ritscher in 2013 showed the doubly exponential nature of Gröbner bases is with respect to the dimension of the ideal, not the number of variables. This inspires us to consider what can be done for Cylindrical Algebraic Decomposition which produces a doubly-exponential number of polynomials of doubly-exponential degree. We review work from ISSAC 2015 which showed the number of polynomials could be restricted to doubly-exponential in the (complex) dimension using McCallum’s theory of reduced projection in the presence of equational constraints. We then discuss preliminary results showing the same for the degree of those polynomials. The results are under primitivity assumptions whose importance we illustrate.
Original languageEnglish
Title of host publicationInternational Congress on Mathematical Software
EditorsGert-Martin Greuel, Thorsten Koch, Peter Paule, Andrew Sommese
Place of PublicationSwitzerland
PublisherSpringer Verlag
Number of pages8
ISBN (Electronic)978-3-319-42432-3
ISBN (Print)978-3-319-42431-6
Publication statusPublished - 6 Jul 2016
EventInternational Congress on Mathematical Software - Berlin, Germany
Duration: 11 Jul 201614 Jul 2016

Publication series

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


ConferenceInternational Congress on Mathematical Software
Abbreviated titleICMS 2016

Bibliographical note

Funded by EU Horizon 2020 FETOPEN-2016-2017-CSA project SC^2 (712689)

The final publication is available at Springer via

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Lecture Notes in Computer Science book series (LNCS, volume 9725)
ISSN 0302-9743


  • Computer algebra
  • Cylindrical algebraic decomposition
  • Equational constraint
  • Gröbner bases
  • Quantifier elimination


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