# Gröbner Bases and Primary Decomposition of Polynomial Ideals

@article{Gianni1988GrbnerBA, title={Gr{\"o}bner Bases and Primary Decomposition of Polynomial Ideals}, author={Patrizia M. Gianni and Barry M. Trager and Gail Zacharias}, journal={J. Symb. Comput.}, year={1988}, volume={6}, pages={149-167} }

## 446 Citations

Primary decomposition of zero-dimensional ideals over arbitrary fields

- Computer Science, Mathematics
- 2014

It is shown that the primary decomposition can be computed in O(nd) field operations and d factorizations of univariate polynomials over the ground field, where n is the number of generators of the polynomial ring and d is the residue class dimension of the ideal.

Properties of Entire Functions Over Polynomial Rings via Gröbner Bases

- MathematicsApplicable Algebra in Engineering, Communication and Computing
- 2003

It is shown that the extension ideals of polynomial prime and primary ideals in the corresponding ring of entire functions remain prime or primary, respectively, and it is proved that a primary decomposition of a polynometric ideal can be extended componentwise to a primary decompposition of the extended ideal.

An algorithm for primary decomposition in polynomial rings over the integers

- Mathematics, Computer Science
- 2010

An algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers using the idea of Shimoyama-Yokoyama and Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo- primary ideals.

New Algorithms for Computing Primary Decomposition of Polynomial Ideals

- Computer Science, MathematicsICMS
- 2010

A new algorithm and its variant for computing a primary decomposition of a polynomial ideal based on the Shimoyama-Yokoyama algorithm can efficiently decompose some ideals which are hard to be decomposed by any of known algorithms.

An algorithm to compute a primary decomposition of modules in polynomial rings over the integers

- Mathematics, Computer Science
- 2014

An algorithm to compute the primary decomposition of a submodule N of the free module ℤ[x1,...,xn]m using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama.

Prime Decompositions of Radicals in Polynomial Rings

- MathematicsJ. Symb. Comput.
- 1994

It is shown that prime decomposition algorithms in R can be lifted to Rx if for every prime ideal P in R univariate polynomials can be factored over the quotient field of the residue class ring R/P.

A survey of primary decomposition using Gröbner bases

- Mathematics
- 1994

A Survey of Primary Decomposition using GrSbner Bases MICHELLE WILSON Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Master of Science We…

On the complexity of computing a Gröbner basis for the radical of a zero dimensional ideal

- MathematicsSTOC '90
- 1990

We show that if a system of polynomials f l , f 2 , . . . ,Jr in n variables with deg(fl) _< d over the rational numbers has only finitely many affine zeros, then, all the affine zeros can be…

Modular absolute decomposition of equidimensional polynomial ideals

- MathematicsArXiv
- 2010

A modular strategy is presented which describes key properties of the absolute primary decomposition of an equidimensional polynomial ideal defined by polynomials with rational coefficients and uses a tricky choice of prime integers to work with.

Gröbner Bases and Primary Decomposition of Modules

- Mathematics, Computer ScienceJ. Symb. Comput.
- 1992

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