### Abstract

Heinzer, Mirbagheri, Ratliff, and Shah investigate parametric decomposition of monomial ideals on a regular sequence of a commutative ring R with identity 1 and prove that if every finite intersection of monomial ideals in R is again a monomial ideal, then each monomial ideal has a unique irredundant parametric decomposition. Sturmfels, Trung, and Vogels prove a similar result without the uniqueness. Bayer, Peeva, and Strumfels study generic monomial ideals, that is monomial ideals in the polynomial ring such that no variable appears with the same nonzero exponent in two different minimal generators, and for these ideals they prove the uniqueness of the irredundant irreducible decompositions and give an algorithm to construct this unique irredundant irreducible decomposition. In this paper, we present three algorithms for finding the unique irredundant irreducible decomposition of any monomial ideal.

Original language | English |
---|---|

Pages (from-to) | 3435-3456 |

Number of pages | 22 |

Journal | Communications in Algebra |

Volume | 30 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2002 Jul 1 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Algorithms on parametric decomposition of monomial ideals.** / Liu, Jung Chen.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 30, no. 7, pp. 3435-3456. https://doi.org/10.1081/AGB-120004497

}

TY - JOUR

T1 - Algorithms on parametric decomposition of monomial ideals

AU - Liu, Jung Chen

PY - 2002/7/1

Y1 - 2002/7/1

N2 - Heinzer, Mirbagheri, Ratliff, and Shah investigate parametric decomposition of monomial ideals on a regular sequence of a commutative ring R with identity 1 and prove that if every finite intersection of monomial ideals in R is again a monomial ideal, then each monomial ideal has a unique irredundant parametric decomposition. Sturmfels, Trung, and Vogels prove a similar result without the uniqueness. Bayer, Peeva, and Strumfels study generic monomial ideals, that is monomial ideals in the polynomial ring such that no variable appears with the same nonzero exponent in two different minimal generators, and for these ideals they prove the uniqueness of the irredundant irreducible decompositions and give an algorithm to construct this unique irredundant irreducible decomposition. In this paper, we present three algorithms for finding the unique irredundant irreducible decomposition of any monomial ideal.

AB - Heinzer, Mirbagheri, Ratliff, and Shah investigate parametric decomposition of monomial ideals on a regular sequence of a commutative ring R with identity 1 and prove that if every finite intersection of monomial ideals in R is again a monomial ideal, then each monomial ideal has a unique irredundant parametric decomposition. Sturmfels, Trung, and Vogels prove a similar result without the uniqueness. Bayer, Peeva, and Strumfels study generic monomial ideals, that is monomial ideals in the polynomial ring such that no variable appears with the same nonzero exponent in two different minimal generators, and for these ideals they prove the uniqueness of the irredundant irreducible decompositions and give an algorithm to construct this unique irredundant irreducible decomposition. In this paper, we present three algorithms for finding the unique irredundant irreducible decomposition of any monomial ideal.

UR - http://www.scopus.com/inward/record.url?scp=0036020736&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036020736&partnerID=8YFLogxK

U2 - 10.1081/AGB-120004497

DO - 10.1081/AGB-120004497

M3 - Article

AN - SCOPUS:0036020736

VL - 30

SP - 3435

EP - 3456

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 7

ER -