Gin and lex of certain monomial ideals

Satoshi Murai; Takayuki Hibi

doi:10.7146/math.scand.a-15000

Gin and lex of certain monomial ideals

Satoshi Murai^*, Takayuki Hibi

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

Let A = K[x₁, . . ., x_n] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg x_i = 1. Given arbitrary integers i and j with 2 ≤ i ≤ n and 3 ≤ j ≤ n, we will construct a monomial ideal I ⊂ A such that (i) β_k(I) < β_k(Gin(I)) for all k < i, (ii) β_i(I) = β_i(Gin(I)), (iii) β_ℓ(Gin(I)) < β_ℓ(Lex(I)) for all ℓ < j and (iv) β_j(Gin(I))=β_j(Lex(I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x₁ > . . . > x_n and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.

Original language	English
Pages (from-to)	76-86
Number of pages	11
Journal	Mathematica Scandinavica
Volume	99
Issue number	1
DOIs	https://doi.org/10.7146/math.scand.a-15000
Publication status	Published - 2006
Externally published	Yes

ASJC Scopus subject areas

Mathematics(all)

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title = "Gin and lex of certain monomial ideals",

abstract = "Let A = K[x1, . . ., xn] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg xi = 1. Given arbitrary integers i and j with 2 ≤ i ≤ n and 3 ≤ j ≤ n, we will construct a monomial ideal I ⊂ A such that (i) βk(I) < βk(Gin(I)) for all k < i, (ii) βi(I) = βi(Gin(I)), (iii) βℓ(Gin(I)) < βℓ(Lex(I)) for all ℓ < j and (iv) βj(Gin(I))=βj(Lex(I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x1 > . . . > xn and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.",

author = "Satoshi Murai and Takayuki Hibi",

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doi = "10.7146/math.scand.a-15000",

language = "English",

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pages = "76--86",

journal = "Mathematica Scandinavica",

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TY - JOUR

T1 - Gin and lex of certain monomial ideals

AU - Murai, Satoshi

AU - Hibi, Takayuki

PY - 2006

Y1 - 2006

N2 - Let A = K[x1, . . ., xn] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg xi = 1. Given arbitrary integers i and j with 2 ≤ i ≤ n and 3 ≤ j ≤ n, we will construct a monomial ideal I ⊂ A such that (i) βk(I) < βk(Gin(I)) for all k < i, (ii) βi(I) = βi(Gin(I)), (iii) βℓ(Gin(I)) < βℓ(Lex(I)) for all ℓ < j and (iv) βj(Gin(I))=βj(Lex(I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x1 > . . . > xn and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.

AB - Let A = K[x1, . . ., xn] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg xi = 1. Given arbitrary integers i and j with 2 ≤ i ≤ n and 3 ≤ j ≤ n, we will construct a monomial ideal I ⊂ A such that (i) βk(I) < βk(Gin(I)) for all k < i, (ii) βi(I) = βi(Gin(I)), (iii) βℓ(Gin(I)) < βℓ(Lex(I)) for all ℓ < j and (iv) βj(Gin(I))=βj(Lex(I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x1 > . . . > xn and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.

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JO - Mathematica Scandinavica

JF - Mathematica Scandinavica

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ER -

Gin and lex of certain monomial ideals

Abstract

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