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
References
SHOWING 1-10 OF 34 REFERENCES
Factorization over finitely generated fields
- MathematicsSYMSAC '81
- 1981
This paper first describes the current methods of factoring polynomials over the integers, and extends them to the integers mod p, and shows that, for a properly specified finitely generated extension of the rationals or the integersmod p, the problem is soluble.
A theoretical basis for the reduction of polynomials to canonical forms
- MathematicsSIGS
- 1976
A characterization theorem is proved for a certain type of bases of polynomial ideals whose usefulness stems from the fact that a number of computability problems in the theory of poynomial ideals is reducible to the construction of bases in this type.
A criterion for detecting unnecessary reductions in the construction of Groebner bases
- MathematicsEUROSAM
- 1979
A new criterion is presented that may be applied in an algorithm for constructing Grobner-bases of polynomial ideals and allows to derive a realistic upper bound for the degrees of the polynomials in the GroBner-Bases computed by the algorithm in the case of poylemials in two variables.
A constructive approach to commutative ring theory
- Mathematics
- 1977
The development of a system in MACSYMA for solving ring problems is described. Fundamental algorithms are given for expressing ideals in canonical form, and concrete examples are demonstrated.
Commutative Algebra, Vohlme L Graduate Texts in Mathematics
- Commutative Algebra, Vohlme L Graduate Texts in Mathematics
- 1975
Ein algorithmus zum Auffinden der Basiselemente des Restklassenringes naeh einem nulldimensionalen Polynomideal
- Ein algorithmus zum Auffinden der Basiselemente des Restklassenringes naeh einem nulldimensionalen Polynomideal
- 1965