scholarly journals On $m$-Closed Graphs

10.37236/4406 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Leila Sharifan ◽  
Masoumeh Javanbakht
Keyword(s):  
Gröbner Basis ◽  
Lexicographic Order ◽  
Primary Decomposition ◽  
Equivalent Condition ◽  
Edge Ideal ◽  
Grobner Basis ◽  
Arbitrary Tree

A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gröbner basis with respect to the lexicographic order induced by $x_1 > \ldots > x_n > y_1> \ldots > y_n$. In this paper, we generalize this notion and study the so called $m$-closed graphs. We find equivalent condition to $3$-closed property of an arbitrary tree $T$. Using it, we classify a class of $3$-closed trees. The primary decomposition of this class of graphs is also studied.

scholarly journals Indispensable Hibi Relations and Gröbner Bases

2015 ◽  
Vol 22 (04) ◽  
pp. 567-580
Author(s):  
Ayesha Asloob Qureshi
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Lexicographic Order ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Ideal Lattices ◽  
The Ideal

In this paper we consider Hibi rings and Rees rings attached to a poset. We classify the ideal lattices of posets whose Hibi relations are indispensable and the ideal lattices of posets whose Hibi relations form a quadratic Gröbner basis with respect to the rank lexicographic order. Similar classifications are obtained for Rees rings of Hibi ideals.

scholarly journals Reverse Lexicographic Gröbner Bases and Strongly Koszul Toric Rings

2016 ◽  
Vol 119 (2) ◽  
pp. 161
Author(s):  
Kazunori Matsuda ◽  
Hidefumi Ohsugi
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Lexicographic Order ◽  
Grobner Bases ◽  
Sufficient Condition ◽  
Grobner Basis ◽  
Toric Ideal ◽  
Necessary And Sufficient ◽  
Partial Extension ◽  
Toric Rings

Restuccia and Rinaldo proved that a standard graded $K$-algebra $K[x_1,\dots,x_n]/I$ is strongly Koszul if the reduced Gröbner basis of $I$ with respect to any reverse lexicographic order is quadratic. In this paper, we give a sufficient condition for a toric ring $K[A]$ to be strongly Koszul in terms of the reverse lexicographic Gröbner bases of its toric ideal $I_A$. This is a partial extension of a result given by Restuccia and Rinaldo. In addition, we show that any strongly Koszul toric ring generated by squarefree monomials is compressed. Using this fact, we show that our sufficient condition for $K[A]$ to be strongly Koszul is both necessary and sufficient when $K[A]$ is generated by squarefree monomials.

scholarly journals Gröbner Bases of Simplicial Toric Ideals

2009 ◽  
Vol 196 ◽  
pp. 67-85 ◽  
Author(s):  
Michael Hellus ◽  
Lê Tûan Hoa ◽  
Jürgen Stückrad
Keyword(s):  
Maximum Degree ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Lexicographic Order ◽  
Grobner Bases ◽  
Toric Ideals ◽  
Grobner Basis

Bounds for the maximum degree of a minimal Gröbner basis of simplicial toric ideals with respect to the reverse lexicographic order are given. These bounds are close to the bound stated in Eisenbud-Goto’s Conjecture on the Castelnuovo-Mumford regularity.

scholarly journals The binomial edge ideal of a pair of graphs

2014 ◽  
Vol 213 ◽  
pp. 105-125 ◽  
Author(s):  
Viviana Ene ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Ayesha Asloob Qureshi
Keyword(s):  
Lower Bound ◽  
Gröbner Basis ◽  
Prime Ideals ◽  
Edge Ideal ◽  
Grobner Basis ◽  
Edge Ideals ◽  
Common Generalization ◽  
Special Cases ◽  
Minimal Prime

AbstractWe introduce a class of ideals generated by a set of 2-minors of an (m×n)-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen–Macaulayness.

scholarly journals The binomial edge ideal of a pair of graphs

2014 ◽  
Vol 213 ◽  
pp. 105-125 ◽  
Author(s):  
Viviana Ene ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Ayesha Asloob Qureshi
Keyword(s):  
Lower Bound ◽  
Gröbner Basis ◽  
Prime Ideals ◽  
Edge Ideal ◽  
Grobner Basis ◽  
Edge Ideals ◽  
Common Generalization ◽  
Special Cases ◽  
Minimal Prime

AbstractWe introduce a class of ideals generated by a set of 2-minors of an (m×n)-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen–Macaulayness.

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.

scholarly journals The Universal Gröbner Basis of a Binomial Edge Ideal

10.37236/5912 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Mourtadha Badiane ◽  
Isaac Burke ◽  
Emil Sköldberg
Keyword(s):  
Complete Graph ◽  
Gröbner Basis ◽  
Basis Set ◽  
Edge Ideal ◽  
Grobner Basis ◽  
Graver Basis ◽  
Underlying Graph ◽  
Universal Gröbner Basis

We show that the universal Gröbner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph.

scholarly journals On the first fall degree of summation polynomials

2019 ◽  
Vol 13 (3-4) ◽  
pp. 229-237
Author(s):  
Stavros Kousidis ◽  
Andreas Wiemers
Keyword(s):  
Elliptic Curve ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Discrete Logarithm Problem ◽  
Grobner Basis ◽  
Weil Descent ◽  
Degree Bound

Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

Sign in / Sign up

Export Citation Format

Share Document