Universal Gröbner Basis for Parametric Polynomial Ideals

2018 ◽  
pp. 191-199
Author(s):  
Amir Hashemi ◽  
Mahdi Dehghani Darmian ◽  
Marzieh Barkhordar
Keyword(s):  
Gröbner Basis ◽  
Grobner Basis ◽  
Polynomial Ideals ◽  
Universal Gröbner Basis

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.

Universal Gröbner basis associated with the maximum flow problem

2012 ◽  
Vol 30 (1) ◽  
pp. 39-50
Author(s):  
Sennosuke Watanabe ◽  
Yoshihide Watanabe ◽  
Daisuke Ikegami
Keyword(s):  
Gröbner Basis ◽  
Flow Problem ◽  
Maximum Flow ◽  
Grobner Basis ◽  
Maximum Flow Problem ◽  
Universal Gröbner Basis

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 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.

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 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.

Enhancing the extended hensel construction by using Gröbner basis

2017 ◽  
Vol 51 (1) ◽  
pp. 26-28
Author(s):  
Tateaki Sasaki ◽  
Daiju Inaba
Keyword(s):  
Gröbner Basis ◽  
Grobner Basis
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