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 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 neural ideals

2018 ◽  
Vol 28 (04) ◽  
pp. 553-571 ◽  
Author(s):  
Rebecca Garcia ◽  
Luis David García Puente ◽  
Ryan Kruse ◽  
Jessica Liu ◽  
Dane Miyata ◽  
...  
Keyword(s):  
Canonical Form ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Boolean Lattice ◽  
Grobner Bases ◽  
Combinatorial Structure ◽  
Grobner Basis ◽  
Neural Codes ◽  
Monomial Order ◽  
Universal Gröbner Basis

The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gröbner basis with respect to that monomial order. How are these two types of generating sets — canonical forms and Gröbner bases — related? Our main result states that if the canonical form of a neural ideal is a Gröbner basis, then it is the universal Gröbner basis (that is, the union of all reduced Gröbner bases). Furthermore, we prove that this situation — when the canonical form is a Gröbner basis — occurs precisely when the universal Gröbner basis contains only pseudo-monomials (certain generalizations of monomials). Our results motivate two questions: (1) When is the canonical form a Gröbner basis? (2) When the universal Gröbner basis of a neural ideal is not a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code? We give partial answers to both questions. Along the way, we develop a representation of pseudo-monomials as hypercubes in a Boolean lattice.

NONCOMMUTATIVE GRÖBNER BASES FOR THE COMMUTATOR IDEAL

2006 ◽  
Vol 16 (01) ◽  
pp. 187-202 ◽  
Author(s):  
SUSAN HERMILLER ◽  
JON McCAMMOND
Keyword(s):  
Associative Algebra ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Free Associative Algebra ◽  
Arbitrary Field ◽  
Grobner Basis ◽  
Commutator Ideal ◽  
Universal Gröbner Basis

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I, and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis.

scholarly journals Strongly Robust Toric Ideals in Codimension 2

2019 ◽  
Vol 10 (1) ◽  
pp. 128-136 ◽  
Author(s):  
Seth Sullivant
Keyword(s):  
Positive Answer ◽  
Homogeneous Ideal ◽  
Toric Ideals ◽  
Grobner Basis ◽  
Toric Ideal ◽  
Generating Set ◽  
Graver Basis ◽  
Minimal Generating Set ◽  
Gale Diagrams ◽  
Universal Gröbner Basis

A homogeneous ideal is robust if its universal Gröbner basis is also a minimal generating set.  For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials.  We characterize the codimension 2  strongly robust toric ideals by their Gale diagrams.  This give a positive answer to a question of Petrovic, Thoma, and Vladoiu in the case of codimension 2 toric ideals.

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 A Hilbert Scheme in Computer Vision

2013 ◽  
Vol 65 (5) ◽  
pp. 961-988 ◽  
Author(s):  
Chris Aholt ◽  
Bernd Sturmfels ◽  
Rekha Thomas
Keyword(s):  
Computer Vision ◽  
Fixed Point ◽  
Hilbert Scheme ◽  
Gröbner Basis ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Grobner Basis ◽  
Multiview Geometry ◽  
Universal Gröbner Basis ◽  
Two Dimensional Images

AbstractMultiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Gröbner basis for the multiview ideal of ngeneric cameras. As the cameras move, the multiview varieties vary in a family of dimension 11n − 15. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme.

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