Gröbner basis and free resolution of the ideal of 2-minors of a 2 ×<i>n</i>matrix of linear forms<sup>*</sup>

Zaare-Nahandi, Rahim; Zaare-Nahandi, Rashid

doi:10.1080/00927870008827098

Cited by 10 publications

(16 citation statements)

References 10 publications

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“…Let H = 4 k=1 H k . Then it is immediate to see [20]). The set H is a Gröbner basis for I λ with respect to the graded reverse lexicographic order > grevlex .…”

Section: Reduced Gröbner Basis Formentioning

confidence: 85%

Eigenschemes and the Jordan canonical form

Abo

Eklund

Kahle³

et al. 2016

Linear Algebra and its Applications

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We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint.Comment: 23 pages; v3: minor corrections, final version as appeared in Linear Algebra App

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“…Let H = 4 k=1 H k . Then it is immediate to see [20]). The set H is a Gröbner basis for I λ with respect to the graded reverse lexicographic order > grevlex .…”

Section: Reduced Gröbner Basis Formentioning

confidence: 85%

Eigenschemes and the Jordan canonical form

Abo

Eklund

Kahle³

et al. 2016

Linear Algebra and its Applications

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“…Generic pluri-circulant matrices naturally arise in the study of the local deÿning equations of generic singularities under certain specializations (see [23,Remark 5.6], [20,21,24]). Some basic properties of the ideals of t-minors of pluri-circulant matrices have been treated in [23].…”

Section: Ideals Of Minors Of Generic Pluri-circulant Matricesmentioning

confidence: 99%

The minimal free resolution of a class of square-free monomial ideals

Zaare-Nahandi

2004

Journal of Pure and Applied Algebra

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For positive integers n; b1 6 b2 6 · · · 6 bn and t 6 n, let It be the transversal monomial ideal generated by square-free monomials yi 1 j 1 yi 2 j 2 · · · yi t jt ;1 6 i1 ¡ i2 ¡ · · · ¡ it 6 n; 1 6 j k 6 bi k ; k = 1; : : : ; t;where yij's are distinct indeterminates. It is observed that the simplicial complex associated to this ideal is pure shellable if and only if b1=· · ·=bn=1, but its Alexander dual is always pure and shellable. The simplicial complex admits some weaker shelling which leads to the computation of its Hilbert series. The main result is the construction of the minimal free resolution for the quotient ring of It. This class of monomial ideals includes the ideals of t-minors of generic pluri-circulant matrices under a change of coordinates. The last family of ideals arise from some specializations of the deÿning ideals of generic singularities of algebraic varieties.

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“…. . , x n ]/I 2 (X) be the determinantal ring of X. Algebraic properties of such determinantal rings R were studied in the literature, see [7], [5] and [22]. The…”

Section: Introductionmentioning

confidence: 99%

“…Concerning the Koszul property, any rational normal scroll is Koszul since it has regularity 1. In fact, any rational normal scroll is also G-quadratic, namely its defining ideal has a quadratic Gröbner basis with respect to a suitable term order; see [22] for a generalization. In this paper, we are able to classify Koszul determinantal rings of 2 × e matrices of linear forms using the Kronecker-Weierstrass theory.…”

Section: Introductionmentioning

confidence: 99%

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Kozul determinantal rings and 2 x e matrices of linear forms

Nguyen¹,

Thieu²,

Vu³

2015

Michigan Math. J.

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Let k be an algebraically closed field of characteristic 0. Let X be a 2 × e matrix of linear forms over a polynomial ring k[x 1 , . . . , x n ] (where e, n ≥ 1). We prove that the determinantal ring R = k[x 1 , . . . , x n ]/I 2 (X) is Koszul if and only if in any Kronecker-Weierstrass normal form of X, the largest length of a nilpotent block is at most twice the smallest length of a scroll block. As an application, we classify rational normal scrolls whose all section rings by natural coordinates are Koszul. This result settles a conjecture due to Conca.

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Gröbner basis and free resolution of the ideal of 2-minors of a 2 ×nmatrix of linear forms^*

Cited by 10 publications

References 10 publications

Eigenschemes and the Jordan canonical form

Eigenschemes and the Jordan canonical form

The minimal free resolution of a class of square-free monomial ideals

Kozul determinantal rings and 2 x e matrices of linear forms

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Gröbner basis and free resolution of the ideal of 2-minors of a 2 ×nmatrix of linear forms*

Cited by 10 publications

References 10 publications

Eigenschemes and the Jordan canonical form

Eigenschemes and the Jordan canonical form

The minimal free resolution of a class of square-free monomial ideals

Kozul determinantal rings and 2 x e matrices of linear forms

Contact Info

Product

Resources

About

Gröbner basis and free resolution of the ideal of 2-minors of a 2 ×nmatrix of linear forms^*