scholarly journals Prestable ideals and Sagbi bases

2005 ◽  
Vol 96 (1) ◽  
pp. 22 ◽  
Author(s):  
Hidefumi Ohsugi ◽  
Takayuki Hibi
Keyword(s):  
Gröbner Basis ◽  
Monomial Ideals ◽  
Rees Algebra ◽  
Grobner Basis ◽  
Toric Ideal

In order to find a reasonable class of squarefree monomial ideals $I$ for which the toric ideal of the Rees algebra of $I$ has a quadratic Gröbner basis, the concept of prestable ideals will be introduced. Prestable ideals arising from finite pure posets together with their application to Sagbi bases will be discussed.

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.

On the Calculation of gl.dim Gℕ(A) and ${\rm gl.dim}\, \widetilde{A}$ by Using Gröbner Bases

2009 ◽  
Vol 16 (02) ◽  
pp. 181-194 ◽  
Author(s):  
Huishi Li
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Graded Algebra ◽  
Rees Algebra ◽  
Grobner Basis

Let [Formula: see text] be a K-algebra defined by a finite Gröbner basis [Formula: see text]. In this paper, it is shown how to use the Ufnarovski graph [Formula: see text] and the graph of n-chains [Formula: see text] to calculate gl.dim Gℕ(A) and [Formula: see text], where Gℕ(A), respectively [Formula: see text], is the associated ℕ-graded algebra of A, respectively the Rees algebra of A with respect to the ℕ-filtration FA of A induced by a weight ℕ-grading filtration of K 〈X1,…, Xn〉.

Recognizing the Semiprimitivity of ℕ-graded Algebras via Gröbner Bases

2015 ◽  
Vol 22 (03) ◽  
pp. 459-468
Author(s):  
Huishi Li
Keyword(s):  
Gröbner Basis ◽  
Graded Algebra ◽  
Rees Algebra ◽  
Grobner Basis ◽  
Graded Algebras ◽  
Ideal Formula ◽  
Monomial Algebra ◽  
Positive Degree ◽  
Monomial Ordering ◽  
The Ideal

Let K〈X〉=K〈X1,…,Xn〉 be the free K-algebra on X={X1,…,Xn} over a field K, which is equipped with a weight ℕ-gradation (i.e., each Xi is assigned a positive degree), and let [Formula: see text] be a finite homogeneous Gröbner basis for the ideal [Formula: see text] of K〈X〉 with respect to some monomial ordering ≺ on K〈X〉. It is shown that if the monomial algebra [Formula: see text] is semiprime, where [Formula: see text] is the set of leading monomials of [Formula: see text] with respect to ≺, then the ℕ-graded algebra A=K〈X〉 /I is semiprimitive in the sense of Jacobson. In the case that [Formula: see text] is a finite nonhomogeneous Gröbner basis with respect to a graded monomial ordering ≺ gr , and the ℕ-filtration FA of the algebra A=K〈X〉 /I induced by the ℕ-grading filtration FK〈X〉 of K〈X〉 is considered, if the monomial algebra [Formula: see text] is semiprime, then it is shown that the associated ℕ-graded algebra G(A) and the Rees algebra à of A determined by FA are all semiprimitive.

scholarly journals On the Grobner Basis of the Toric Ideal for 3 X n- Contingency Tables

2019 ◽  
pp. 1362-1366
Author(s):  
Hussein S. Mohammed Hussein ◽  
Abdulrahman H. Majeed
Keyword(s):  
Gröbner Basis ◽  
Contingency Tables ◽  
Grobner Basis ◽  
Toric Ideal ◽  
Markov Basis ◽  
Universal Gröbner Basis

In this paper, The Grobner basis of the Toric Ideal for - contingency tables related with the Markov basis B introduced by Hussein S. MH, Abdulrahman H. M in 2018 is found. Also, the Grobner basis is a reduced and universal Grobner basis are shown.

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.

scholarly journals On the relation between the MXL family of algorithms and Gröbner basis algorithms

2012 ◽  
Vol 47 (8) ◽  
pp. 926-941 ◽  
Author(s):  
Martin R. Albrecht ◽  
Carlos Cid ◽  
Jean-Charles Faugère ◽  
Ludovic Perret
Keyword(s):  
Gröbner Basis ◽  
Grobner Basis
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