The positivity of the first coefficients of normal Hilbert polynomials

Gotō, Shirō; Hong, Jooyoun; Mandal, Mousumi

doi:10.1090/s0002-9939-2010-10710-4

Cited by 8 publications

(13 citation statements)

References 9 publications

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“…The next lemma is crucial for applying induction on the dimension of the ring. It is proved in [20] and [10].…”

Section: The Positivity Conjecturementioning

confidence: 86%

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Normal Hilbert Polynomials : A survey

Mandal¹,

Masuti²,

Verma³

2012

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We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in Cohen-Macaulay analytically unramified local rings. We also discuss recent works of Goto-Hong-Mandal and Mandal-Singh-Verma concerning the positivity of the first coefficient of the normal Hilbert polynomial in unmixed analytically unramified local rings. Results of Moralés and Villarreal linking normal Hilbert polynomial of monomial ideals with Ehrhart polynomials of polytopes are also presented. Contents 14 5. Study of e 3 (I) 19 6. Normal Hilbert Polynomials in two dimensional regular local rings 25 7. The normal Hilbert polynomial of a monomial ideal 30 References 33

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“…The next lemma is crucial for applying induction on the dimension of the ring. It is proved in [20] and [10].…”

Section: The Positivity Conjecturementioning

confidence: 86%

“…In [10] the solution to the Positivity Conjecture has been given for analytically unramified unmixed local rings. In order to prove the Theorem we need a lemma.…”

Section: The Positivity Conjecturementioning

confidence: 99%

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Normal Hilbert Polynomials : A survey

Mandal¹,

Masuti²,

Verma³

2012

Preprint

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“…As A is analytically unramified the integral closure filtration of a is a-stable, see [34] (also see [13, 9.1.2]). The integral closure filtration of m-primary ideals has inspired plenty of research, see for instance [39,Appendix 5], [35], [12], [16], [17], [20], [21] [7], [8], [5] and [38]. (ii) If A contains a field of characteristic p > 0 then the tight closure filtration of a has been of some interest recently, see [6].…”

Section: Introductionmentioning

confidence: 99%

Itoh's conjecture for normal ideals

Puthenpurakal¹

2022

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Let (A, m) be an analytically unramified Cohen-Macaulay local ring and let a be an m-primary ideal in A. If I is an ideal in A then let I * be the integral closure of) * be the associated graded ring of the integral closure filtration of a. Itoh conjectured that if e a * 3 (A) = 0 and A is Gorenstein then Ga(A) * is Cohen-Macaulay. In this paper we prove an important case of Itoh's conjecture: we show that if A is Cohen-Macaulay and if a is normal (i.e., a n is integrally closed for all n ≥ 1) with e a 3 (A) = 0 then Ga(A) is Cohen-Macaulay.

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“…Marley [9] showed that if I is an I-admissible filtration then the Hilbert function H I (n) = λ(R/I n ), where λ denotes length as R-module, coincides with a polynomial P I (x) ∈ Q[x] of degree d for large n. This polynomial is The Chern number has traditionally been studied in Cohen-Macaulay local rings. The recent solutions by Goto et al [2,3,4] of conjectures of Vasconcelos [13] for the Chern numbers of parameter ideals and integral closure filtrations require their understanding in arbitrary local rings. Therefore it is useful to have versions of important formulas for the Chern numbers in general.…”

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confidence: 99%