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.

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M3 - Conference contribution

AN - SCOPUS:0028333510

SN - 0898713293

T3 - Proceedings of the Annual ACM SIAM Symposium on Discrete Algorithms

SP - 280

EP - 289

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

PB - Publ by ACM

T2 - Proceedings of the Fifth Annual SIAM Symposium on Discrete Algorithms

Y2 - 23 January 1994 through 25 January 1994

ER -