Complexity of resolvent resolved

Giovanni Gallo, Bhubaneswar Mishra

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

Abstract

The concept of a resolvent of a prime ideal was originally introduced by J. F. Ritt along with the notion of a characteristic set. The motivation for studying resolvents comes from its connections with the birational isomorphisms that describe structures of irreducible algebraic varieties by means of an equivalent hypersurface and a one-to-one rational map. As a result, these ideas have a wide range of applications in such areas as solid modeling, computer-aided design and manufacturing. An algorithm to compute the resolvent by means of characteristic sets was first proposed by Ritt. This and some related algorithms have resurfaced as interest in resolvent structures have grown, spurred by its applicability. Unfortunately, the algebraic complexity of the resolvent and the computational complexity of the associated algorithms have never been explicitly explored. In this paper, we give single exponential upper and lower bounds for the degrees of the resolvent and its associated parametrizing polynomials. We also show that the resolvent can be computed deterministically in single exponential sequential and polynomial parallel time complexity. All previous algorithms for resolvent had relied on a random choice of certain extraneous parameters.

Original languageEnglish (US)
Title of host publicationProceedings of the Annual ACM SIAM Symposium on Discrete Algorithms
PublisherPubl by ACM
Pages280-289
Number of pages10
ISBN (Print)0898713293
StatePublished - 1994
EventProceedings of the Fifth Annual SIAM Symposium on Discrete Algorithms - Arlington, VA, USA
Duration: Jan 23 1994Jan 25 1994

Other

OtherProceedings of the Fifth Annual SIAM Symposium on Discrete Algorithms
CityArlington, VA, USA
Period1/23/941/25/94

Fingerprint

Resolvent
Polynomials
Characteristic Set
Computer aided manufacturing
Computational complexity
Computer aided design
Algebraic Complexity
Computer Aided Manufacturing
Solid Modeling
Rational Maps
Associated Polynomials
Algebraic Variety
Computer simulation
Computer-aided Design
Prime Ideal
Time Complexity
Hypersurface
Upper and Lower Bounds
Isomorphism
Computational Complexity

ASJC Scopus subject areas

  • Chemical Health and Safety
  • Software
  • Safety, Risk, Reliability and Quality
  • Discrete Mathematics and Combinatorics

Cite this

Gallo, G., & Mishra, B. (1994). Complexity of resolvent resolved. In Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms (pp. 280-289). Publ by ACM.

Complexity of resolvent resolved. / Gallo, Giovanni; Mishra, Bhubaneswar.

Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms. Publ by ACM, 1994. p. 280-289.

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

Gallo, G & Mishra, B 1994, Complexity of resolvent resolved. in Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms. Publ by ACM, pp. 280-289, Proceedings of the Fifth Annual SIAM Symposium on Discrete Algorithms, Arlington, VA, USA, 1/23/94.
Gallo G, Mishra B. Complexity of resolvent resolved. In Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms. Publ by ACM. 1994. p. 280-289
Gallo, Giovanni ; Mishra, Bhubaneswar. / Complexity of resolvent resolved. Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms. Publ by ACM, 1994. pp. 280-289
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