scholarly journals Weakly Closed Graphs and F-Purity of Binomial Edge Ideals

2018 ◽  
Vol 25 (04) ◽  
pp. 567-578
Author(s):  
Kazunori Matsuda
Keyword(s):  
Polynomial Ring ◽  
Quotient Ring ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Edge Ideal ◽  
Closed Graph ◽  
Edge Ideals ◽  
Weakly Closed

Herzog, Hibi, Hreindóttir et al. introduced the class of closed graphs, and they proved that the binomial edge ideal JG of a graph G has quadratic Gröbner bases if G is closed. In this paper, we introduce the class of weakly closed graphs as a generalization of the closed graph, and we prove that the quotient ring S/JG of the polynomial ring [Formula: see text] with K a field and [Formula: see text] is F-pure if G is weakly closed. This fact is a generalization of Ohtani’s theorem.

scholarly journals Binomial Edge Ideals with Quadratic Gröbner Bases

10.37236/698 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Marilena Crupi ◽  
Giancarlo Rinaldo
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Edge Ideal ◽  
Grobner Basis ◽  
Edge Ideals ◽  
Term Order ◽  
Vertex Set

We prove that a binomial edge ideal of a graph $G$ has a quadratic Gröbner basis with respect to some term order if and only if the graph $G$ is closed with respect to a given labelling of the vertices. We also state some criteria for the closedness of a graph $G$ that do not depend on the labelling of its vertex set.

Border bases and kernels of homomorphisms and of derivations

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Janusz Zieliński
Keyword(s):  
Polynomial Ring ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
The Ideal

AbstractBorder bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.

scholarly journals Binomial Edge Ideals of Graphs

10.37236/2349 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Dariush Kiani ◽  
Sara Saeedi
Keyword(s):  
Upper Bound ◽  
Betti Number ◽  
Complete Graphs ◽  
Edge Ideal ◽  
Closed Graph ◽  
Edge Ideals ◽  
Linear Resolution

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of a closed graph.

scholarly journals Upper bounds for the regularity of symbolic powers of certain classes of edge ideals

2021 ◽  
Author(s):  
Arvind Kumar ◽  
S. Selvaraja
Keyword(s):  
Polynomial Ring ◽  
Upper Bounds ◽  
Simple Graph ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Symbolic Powers

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote the corresponding edge ideal in a polynomial ring over a field [Formula: see text]. In this paper, we obtain upper bounds for the Castelnuovo–Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.

On Finite Noncommutative Gröbner Bases

2020 ◽  
Vol 27 (03) ◽  
pp. 381-388
Author(s):  
Yatma Diop ◽  
Djiby Sow
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Ideal Formula ◽  
Monomial Ordering ◽  
Noncommutative Polynomial

It is well known that in the noncommutative polynomial ring in serveral variables Buchberger’s algorithm does not always terminate. Thus, it is important to characterize noncommutative ideals that admit a finite Gröbner basis. In this context, Eisenbud, Peeva and Sturmfels defined a map γ from the noncommutative polynomial ring k⟨X1, …, Xn⟩ to the commutative one k[x1, …, xn] and proved that any ideal [Formula: see text] of k⟨X1, …, Xn⟩, written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn], amits a finite Gröbner basis with respect to a special monomial ordering on k⟨X1, …, Xn⟩. In this work, we approach the opposite problem. We prove that under some conditions, any ideal [Formula: see text] of k⟨X1, …, Xn⟩ admitting a finite Gröbner basis can be written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn].

scholarly journals POWERS OF BINOMIAL EDGE IDEALS WITH QUADRATIC GRÖBNER BASES

2021 ◽  
pp. 1-23
Author(s):  
VIVIANA ENE ◽  
GIANCARLO RINALDO ◽  
NAOKI TERAI
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Edge Ideals ◽  
Block Graphs

Abstract We study powers of binomial edge ideals associated with closed and block graphs.

scholarly journals Chinese remainder theorem secret sharing in multivariate polynomials

2019 ◽  
pp. 129-133
Author(s):  
Gennadii V. Matveev
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Secret Sharing ◽  
Chinese Remainder Theorem ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Multivariate Polynomials ◽  
Multivariate Polynomial ◽  
Ring Of Integers

This paper deals with a generalization of the secret sharing using Chinese remainder theorem over the integers to multivariate polynomials over a finite field. We work with the ideals and their Gröbner bases instead of integer moduli. Therefore, the proposed method is called GB secret sharing. It was initially presented in our previous paper. Now we prove that any threshold structure has ideal GB realization. In a generic threshold modular scheme in ring of integers the sizes of the share space and the secret space are not equal. So, the scheme is not ideal and our generalization of modular secret sharing to the multivariate polynomial ring is more secure.

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