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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 13P: Computational aspects of commutative algebra

Introduction

History

Applications and related fields

The computational tools described on this page are used in other areas, particularly algebraic geometry and its subfields (e.g. one can compute the envelope of a curve). Roughly speaking we have included here the comments which are best exemplified with varieties of dimension zero (finite sets of points) and in section 14 the comments which involve more geometry than computation.

Computation in polynomial rings overlaps 12F: Field extensions (Galois theory). In particular, look there for computational questions involving the factorization of univariate polynomials.

There have been a number of applications to topics in robotics and the motions of linked systems.

See Also 68W30

Subfields

• 13P05: Polynomials, factorization, See also 12Y05
• 13P10: Polynomial ideals, Gröbner bases
• 13P99: None of the above but in this section

Parent field: 13: Commutative Rings and Algebras

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Software and tables

Other web sites with this focus

Selected topics at this site

• Announcement: DoCon software (Algebraic Domain Constructor)
• Announcement: Singular computer algebra system for polynomial computations.
• References describing Gröbner bases
• A general pointer to web site discussing Gröbner bases etc.
• Gröbner bases (bases for ideals in polynomial rings which permit rapid computations): citations + pointers for general descriptions.
• Pointers to Gröbner basis source code
• Citations for computing prime ideal decomposition.
• Why Gröbner bases grow so nastily; any way around that?
• Example of intermediate swell of Gröbner bases in elimination theory
• Using Gröbner bases to find closed-form solutions to multivariate recurrence relations and difference equations.
• Using Gröbner bases to determine the image of a polynomial map.
• Obtaining the Smith normal form for matrices (or modules) over a PID.
• Factoring the resultant of two 1-variable polynomials.
• Citations for efficient multivariate resultants.
• Use of resultants as an efficient alternative(!) to Gröbner bases.
• Example of elimination (implicitization of parameterized curve) using inexact coefficients.
• Use Gröbner bases or resultants on polynomials with inexact coefficients?
• Using elimination to help analyze a curve in R^3 specified by two polynomial equations
• Typical geometric enumeration problem: compute all arrangements of 3 lines tangent to 3 given balls and perpendicular to each other.
• Typical elimination problem: compute the (8) points of intersection of 3 perpendicular cylinders centered at the origin.
• Using elimination to describe the (40) points in a particular variety of dimension zero
• Recent progress on solving polynomial systems, and multidimensional resultants; lit review. [J. Maurice Rojas]
• Are Gröbner bases better than resultants?
• Does a set of real polynomials have a real solution? Tarski's Elimination of Quantifiers.
• Is the real locus of an algebraic variety non-empty? (calculable)
• Using the Chinese Remainder Theorem: eliminating (say) modulo many small primes and lifting to get a rational elimination.