Computations with Algebraic Curves

@inproceedings{Abhyankar1988ComputationsWA,
title={Computations with Algebraic Curves},
author={Shreeram S. Abhyankar and Chandrajit L. Bajaj},
booktitle={ISSAC},
year={1988}
}
• Published in ISSAC 4 July 1988
• Mathematics, Computer Science
We present a variety of computational techniques dealing with algebraic curves both in the plane and in space. Our main results are polynomial time algorithms (1) to compute the genus of plane algebraic curves, (2) to compute the rational parametric equations for implicitly defined rational plane algebraic curves of arbitrary degree, (3) to compute birational mappings between points on irreducible space curves and points on projected plane curves and thereby to compute the genus and rational…

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References

SHOWING 1-10 OF 71 REFERENCES

The computerization of algebraic geometry

The proper mathematical foundations for the treatment of algebraic functions are presented, and the formalism leads directly to the requirement for algorithms to find the genus of an algebraic curve, and to discover what function is associated with a given divisor.

Precise Implementation of Cad Primitives using Rational Parameterizations of Standard Surfaces

• Mathematics
• 1984
We discuss the problem of computing the intersection curve of two algebraic surfaces, each of which possesses rational parameterization. The special case where the two surfaces are quadric is

Algebraic curves

This introduction to algebraic geometry examines how the more recent abstract concepts relate to traditional analytical and geometrical problems. The presentation is kept as elementary as A linear

Implicit representation of parametric curves and surfaces

• Mathematics, Computer Science
Comput. Vis. Graph. Image Process.
• 1984

Generalized Polynomial Remainder Sequences

• R. Loos
• Mathematics, Computer Science
• 1983
Habicht’s subresultant theorem allows new and simple proofs of many results and algorithms found in different ways in Computer algebra.

Tracing surface intersections

• Mathematics
Comput. Aided Geom. Des.
• 1988