Generalized minimum distance functions and algebraic invariants of Geramita ideals

Cooper, Susan; Seceleanu, Alexandra; Tohǎneanu, Ştefan; Pinto, Maria Vaz; Villarreal, Rafael H.

doi:10.1016/j.aam.2019.101940

Cited by 26 publications

(22 citation statements)

References 29 publications

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“…By convention, we set α(0) := 0. Part (d) of the next result was shown in Proposition 4.2 in [1] for unmixed graded ideals. The next result gives a formula for the v-number of any graded ideal.…”

Section: Introductionmentioning

confidence: 64%

“…The set of associated primes of S/I is denoted by Ass(I), and the set of maximal elements of Ass(I) with respect to inclusion is denoted by Max(I). The v-number of I, denoted v(I), is the following invariant of I that was introduced in [1] to study the asymptotic behavior of the minimum distance of projective Reed-Muller-type codes, Corollary 4.7 in [1]: v(I) := min{d ≥ 0 | ∃ f ∈ S d and p ∈ Ass(I) with (I : f ) = p}.…”

Section: Introductionmentioning

confidence: 99%

“…For certain classes of graded ideals, v(I) is a lower bound for reg(S/I), the regularity of the quotient ring S/I (Definition 1); see [1,4,5]. There are examples of ideals where v(I) > reg(S/I) [4].…”

Section: Introductionmentioning

confidence: 99%

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Induced Matchings and the v-Number of Graded Ideals

2021

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We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of I(G) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W2-graphs.

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Section: Introductionmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 99%

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Induced Matchings and the v-Number of Graded Ideals

2021

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“…The v-number of I was introduced as an invariant of the graded ideal I, in [8], in the study of Reed-Muller-type codes. This invariant of I helps us understand the behaviour of the generalized minimum distance function δ I of I, in the said context.…”

Section: Introductionmentioning

confidence: 99%

“…This invariant of I helps us understand the behaviour of the generalized minimum distance function δ I of I, in the said context. See [8], [16], [21], [23], for further details on this.…”

Section: Introductionmentioning

confidence: 99%

The $\mathrm{v}$-number of Monomial Ideals

Kamalesh¹,

Sengupta²

2021

Preprint

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We generalize some results of v-number for arbitrary monomial ideals by showing that the v-number of an arbitrary monomial ideal is the same as the v-number of its polarization. We prove that the vnumber v(I(G)) of the edge ideal I(G), the induced matching number im(G) and the regularity reg, where G is either a bipartite graph, or a (C 4 , C 5 )free vertex decomposable graph, or a whisker graph. There is an open problem in [16], whether v(I) ≤ reg(R/I) + 1 for any square-free monomial ideal I. We show that v(I(G)) > reg(R/I(G)) + 1, for a disconnected graph G. We derive some inequalities of v-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v(I(G)) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that reg(R/I(G)) can be arbitrarily larger than v(I(G)). Also, we try to see how the v-number is related to the Cohen-Macaulay property of square-free monomial ideals.

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Geometry of the minimum distance

Pawlina,

Tohǎneanu

2024

AAECC

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Let $${{\mathbb {K}}}$$ K be any field, let $$X\subset {\mathbb P}^{k-1}$$ X ⊂ P k - 1 be a set of $$n$$ n distinct $${{\mathbb {K}}}$$ K -rational points, and let $$a\ge 1$$ a ≥ 1 be an integer. In this paper we find lower bounds for the minimum distance $$d(X)_a$$ d ( X ) a of the evaluation code of order $$a$$ a associated to $$X$$ X . The first results use $$\alpha (X)$$ α ( X ) , the initial degree of the defining ideal of $$X$$ X , and the bounds are true for any set $$X$$ X . In another result we use $$s(X)$$ s ( X ) , the minimum socle degree, to find a lower bound for the case when $$X$$ X is in general linear position. In both situations we improve and generalize known results.

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Generalized minimum distance functions and algebraic invariants of Geramita ideals

Cited by 26 publications

References 29 publications

Induced Matchings and the v-Number of Graded Ideals

Induced Matchings and the v-Number of Graded Ideals

The $\mathrm{v}$-number of Monomial Ideals

Geometry of the minimum distance

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