The multiplicity and the number of generators of an integrally closed ideal

Dao, Hailong; Smirnov, Ilya

doi:10.5427/jsing.2019.19e

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…The proof of this theorem was given by Hailong Dao at MathOverflow [12]. We present it here for the convenience of the reader.…”

Section: Theorem 4 ([12]mentioning

confidence: 99%

PIT for depth-$4$ circuits and Sylvester-Gallai conjecture for polynomials

Milovanov¹

2019

Preprint

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This text is a development of preprint [11]. We present an approach for devising a deterministic polynomial time blackbox identity testing (PIT) algorithm for depth-4 circuits with bounded top fanin. This approach is similar to Kayal-Shubhangi [15] approach for depth-3 circuits. Kayal and Shubhangi based their algorithm on Sylvester-Gallai-type theorem about linear polynomials. We show how it is possible to generalize this approach to depth-4 circuits. However we failed to implement this plan completely. We succeeded to construct a polynomial time deterministic algorithm for depth-4 circuits with bounded top fanin and its correctness requires a hypothesis. Also we present a polynomial-time (unconditional) algorithm for some subclass of depth-4 circuits with bounded top fanin.

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“…The proof of this theorem was given by Hailong Dao at MathOverflow [12]. We present it here for the convenience of the reader.…”

Section: Theorem 4 ([12]mentioning

confidence: 99%

PIT for depth-$4$ circuits and Sylvester-Gallai conjecture for polynomials

Milovanov¹

2019

Preprint

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“…In this brief section we explore another set of related conjectures originating in a paper of Dao and Smirnov ([3]). In [3,Theorem 3.1] dimension two, this happens to be, in fact, an equality µ(I) − 1 = e 1 (m | I). However, the proof of [3, Theorem 3.1] shows that in dimension at least three the displayed inequality is never an equality.…”

Section: Number Of Generatorsmentioning

confidence: 99%

A generalization of an inequality of Lech relating multiplicity and colength*

Huneke

Smirnov

Validashti

2019

Communications in Algebra

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We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength, and prove one of these generalizations involving the multiplicity of the maximal ideal times the finite colength ideal. We also propose a Lech-type formula that relates multiplicity and the number of generators. We prove the conjecture in dimension three and establish a weaker result in full generality.

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“…He presented an inequality between µ(I), the multiplicity e(I) of I and Loewy length ℓℓ(I). Dao and Smirnov [2] proved that the Question 5.1 holds true if A is a twodimensional analytically unramified and the maximal ideal m is a p g -ideal (or, equivalently, the Rees algebra R(m) is normal and Cohen-Macaulay).…”

Section: Introductionmentioning

confidence: 99%

The strong Rees property of powers of the maximal ideal and Takahashi–Dao's question

2021

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Dedicated to Craig Huneke on the occasion of his 65th Birthday.Abstract. In this paper, we introduce the notion of the strong Rees property (SRP) for m-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal m has its property if the associated graded ring G of m satisfies depth G ≥ 2. As its application, we characterize two-dimensional excellent normal local domains so that m is a pg-ideal.Finally we ask what m-primary ideals have SRP and state a conjecture which characterizes the case when m n are the only ideals which have SRP.Question 1.2. Assume that (A, m) is a two-dimensional excellent normal local domain. If an inequality (µ(I) − 1) · ℓℓ(I) ≥ e(I) holds for any m-primary integrally closed ideal I, then is m a p g -ideal?It turns out that this question is related to the "strong Rees property" of powers of the maximal ideal. The main result in this paper is the following theorem. Theorem 3.2. Let (A, m) be a Noetherian local ring. Assume that depth A ≥ 2 and H 1 M (G) has finite length, where G = G(m) = ⊕ n≥0 m n /m n+1 . If m ℓ is Ratliff-Rush closed, then m ℓ has the strong Rees property.By Remark 2.5, any power m ℓ is Ratliff-Rush closed whenever depth G ≥ 1. Hence we have the following corollary.

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The multiplicity and the number of generators of an integrally closed ideal

Cited by 4 publications

References 14 publications

PIT for depth-$4$ circuits and Sylvester-Gallai conjecture for polynomials

PIT for depth-$4$ circuits and Sylvester-Gallai conjecture for polynomials

A generalization of an inequality of Lech relating multiplicity and colength*

The strong Rees property of powers of the maximal ideal and Takahashi–Dao's question

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