### Abstract

A Gotzmann monomial ideal of a polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. Let R = K[x_{1},...,x_{n}] denote the polynomial ring in n variables over a field K and M^{d} the set of monomials of R of degree d. A subset V ⊂ M^{d} is said to be a Gotzmann subset if the ideal generated by V is a Gotzmann monomial ideal. In the present paper, we find all integers a > 0 such that every Gotzmann subset V ⊂ M^{d} with |V| = a is lexsegment (up to the permutations of the variables). In addition, we classify all Gotzmann subsets of K[x_{1}, x_{2}, x_{3}].

Original language | English |
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Pages (from-to) | 843-852 |

Number of pages | 10 |

Journal | Illinois Journal of Mathematics |

Volume | 51 |

Issue number | 3 |

Publication status | Published - 2007 Sep 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Gotzmann monomial ideals.** / Murai, Satoshi.

Research output: Contribution to journal › Article

*Illinois Journal of Mathematics*, vol. 51, no. 3, pp. 843-852.

}

TY - JOUR

T1 - Gotzmann monomial ideals

AU - Murai, Satoshi

PY - 2007/9/1

Y1 - 2007/9/1

N2 - A Gotzmann monomial ideal of a polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. Let R = K[x1,...,xn] denote the polynomial ring in n variables over a field K and Md the set of monomials of R of degree d. A subset V ⊂ Md is said to be a Gotzmann subset if the ideal generated by V is a Gotzmann monomial ideal. In the present paper, we find all integers a > 0 such that every Gotzmann subset V ⊂ Md with |V| = a is lexsegment (up to the permutations of the variables). In addition, we classify all Gotzmann subsets of K[x1, x2, x3].

AB - A Gotzmann monomial ideal of a polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. Let R = K[x1,...,xn] denote the polynomial ring in n variables over a field K and Md the set of monomials of R of degree d. A subset V ⊂ Md is said to be a Gotzmann subset if the ideal generated by V is a Gotzmann monomial ideal. In the present paper, we find all integers a > 0 such that every Gotzmann subset V ⊂ Md with |V| = a is lexsegment (up to the permutations of the variables). In addition, we classify all Gotzmann subsets of K[x1, x2, x3].

UR - http://www.scopus.com/inward/record.url?scp=46849105869&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=46849105869&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:46849105869

VL - 51

SP - 843

EP - 852

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

SN - 0019-2082

IS - 3

ER -