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 $m$-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 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 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 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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