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 Complexity bounds on Semaev’s naive index calculus method for ECDLP

2020 ◽  
Vol 14 (1) ◽  
pp. 460-485
Author(s):  
Kazuhiro Yokoyama ◽  
Masaya Yasuda ◽  
Yasushi Takahashi ◽  
Jun Kogure
Keyword(s):  
Lower Bound ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Grobner Basis ◽  
Step Method ◽  
Complexity Bounds ◽  
Index Calculus ◽  
Basis Computation ◽  
The Ideal

AbstractSince Semaev introduced summation polynomials in 2004, a number of studies have been devoted to improving the index calculus method for solving the elliptic curve discrete logarithm problem (ECDLP) with better complexity than generic methods such as Pollard’s rho method and the baby-step and giant-step method (BSGS). In this paper, we provide a deep analysis of Gröbner basis computation for solving polynomial systems appearing in the point decomposition problem (PDP) in Semaev’s naive index calculus method. Our analysis relies on linear algebra under simple statistical assumptions on summation polynomials. We show that the ideal derived from PDP has a special structure and Gröbner basis computation for the ideal is regarded as an extension of the extended Euclidean algorithm. This enables us to obtain a lower bound on the cost of Gröbner basis computation. With the lower bound, we prove that the naive index calculus method cannot be more efficient than generic methods.

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.

Balancing sets of vectors

2010 ◽  
Vol 47 (3) ◽  
pp. 333-349 ◽  
Author(s):  
Gábor Hegedűs
Keyword(s):  
Lower Bound ◽  
Linear Algebra ◽  
Scalar Product ◽  
Gröbner Basis ◽  
Prime Factor ◽  
Grobner Basis ◽  
Arbitrary Integer ◽  
Usual Scalar ◽  
Usual Scalar Product ◽  
Minimal Integer

Let n be an arbitrary integer, let p be a prime factor of n . Denote by ω1 the pth primitive unity root, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\omega _1 : = e^{\tfrac{{2\pi i}} {p}}$$ \end{document}.Define ωi ≔ ω1i for 0 ≦ i ≦ p − 1 and B ≔ {1, ω1 , …, ωp −1 } n ⊆ ℂ n .Denote by K ( n; p ) the minimum k for which there exist vectors ν1 , …, νk ∈ B such that for any vector w ∈ B , there is an i , 1 ≦ i ≦ k , such that νi · w = 0, where ν · w is the usual scalar product of ν and w .Gröbner basis methods and linear algebra proof gives the lower bound K ( n; p ) ≧ n ( p − 1).Galvin posed the following problem: Let m = m ( n ) denote the minimal integer such that there exists subsets A1 , …, Am of {1, …, 4 n } with | Ai | = 2 n for each 1 ≦ i ≦ n , such that for any subset B ⊆ [4 n ] with 2 n elements there is at least one i , 1 ≦ i ≦ m , with Ai ∩ B having n elements. We obtain here the result m ( p ) ≧ p in the case of p > 3 primes.

scholarly journals A New Approach to Nonlinear Invariants for Hybrid Systems Based on the Citing Instances Method

Information ◽  
2020 ◽  
Vol 11 (5) ◽  
pp. 246
Author(s):  
Honghui He ◽  
Jinzhao Wu
Keyword(s):  
Hybrid Systems ◽  
Gröbner Basis ◽  
State Variables ◽  
Grobner Basis ◽  
Constraint Equations ◽  
Free Variables ◽  
Special Cases ◽  
Finite Set ◽  
Algebraic Constraints ◽  
System Variables

In generating invariants for hybrid systems, a main source of intractability is that transition relations are first-order assertions over current-state variables and next-state variables, which doubles the number of system variables and introduces many more free variables. The more variables, the less tractability and, hence, solving the algebraic constraints on complete inductive conditions by a comprehensive Gröbner basis is very expensive. To address this issue, this paper presents a new, complete method, called the Citing Instances Method (CIM), which can eliminate the free variables and directly solve for the complete inductive conditions. An instance means the verification of a proposition after instantiating free variables to numbers. A lattice array is a key notion in this paper, which is essentially a finite set of instances. Verifying that a proposition holds over a Lattice Array suffices to prove that the proposition holds in general; this interesting feature inspires us to present CIM. On one hand, instead of computing a comprehensive Gröbner basis, CIM uses a Lattice Array to generate the constraints in parallel. On the other hand, we can make a clever use of the parallelism of CIM to start with some constraint equations which can be solved easily, in order to determine some parameters in an early state. These solved parameters benefit the solution of the rest of the constraint equations; this process is similar to the domino effect. Therefore, the constraint-solving tractability of the proposed method is strong. We show that some existing approaches are only special cases of our method. Moreover, it turns out CIM is more efficient than existing approaches under parallel circumstances. Some examples are presented to illustrate the practicality of our method.

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.

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