### 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 language | English (US) |
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Title of host publication | Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms |

Publisher | Publ by ACM |

Pages | 280-289 |

Number of pages | 10 |

ISBN (Print) | 0898713293 |

State | Published - 1994 |

Event | Proceedings of the Fifth Annual SIAM Symposium on Discrete Algorithms - Arlington, VA, USA Duration: Jan 23 1994 → Jan 25 1994 |

### Other

Other | Proceedings of the Fifth Annual SIAM Symposium on Discrete Algorithms |
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City | Arlington, VA, USA |

Period | 1/23/94 → 1/25/94 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms*(pp. 280-289). Publ by ACM.

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - Complexity of resolvent resolved

AU - Gallo, Giovanni

AU - Mishra, Bhubaneswar

PY - 1994

Y1 - 1994

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0028333510&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028333510&partnerID=8YFLogxK

M3 - Conference contribution

SN - 0898713293

SP - 280

EP - 289

BT - Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms

PB - Publ by ACM

ER -