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 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.

scholarly journals Gröbner bases and products of coefficient rings

2002 ◽  
Vol 65 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Graham H. Norton ◽  
Ana Sӑlӑgean
Keyword(s):  
Gröbner Basis ◽  
Chinese Remainder Theorem ◽  
Gröbner Bases ◽  
Commutative Rings ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Know How ◽  
Elementary Method ◽  
Coefficient Ring ◽  
Finite Direct Product

Suppose that A is a finite direct product of commutative rings. We show from first principles that a Gröbner basis for an ideal of A[x1,…,xn] can be easily obtained by ‘joining’ Gröbner bases of the projected ideals with coefficients in the factors of A (which can themselves be obtained in parallel). Similarly for strong Gröbner bases. This gives an elementary method of constructing a (strong) Gröbner basis when the Chinese Remainder Theorem applies to the coefficient ring and we know how to compute (strong) Gröbner bases in each factor.

scholarly journals Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

2014 ◽  
Vol 24 (08) ◽  
pp. 1157-1182 ◽  
Author(s):  
Roberto La Scala
Keyword(s):  
Associative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Free Associative Algebra ◽  
Commutative Algebras ◽  
Skew Polynomial Rings ◽  
Experimental Implementation ◽  
Buchberger Algorithm ◽  
Skew Polynomial

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

scholarly journals Indispensable Hibi Relations and Gröbner Bases

2015 ◽  
Vol 22 (04) ◽  
pp. 567-580
Author(s):  
Ayesha Asloob Qureshi
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Lexicographic Order ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Ideal Lattices ◽  
The Ideal

In this paper we consider Hibi rings and Rees rings attached to a poset. We classify the ideal lattices of posets whose Hibi relations are indispensable and the ideal lattices of posets whose Hibi relations form a quadratic Gröbner basis with respect to the rank lexicographic order. Similar classifications are obtained for Rees rings of Hibi ideals.

scholarly journals Subalgebra Analogue to standard basis for ideal

2011 ◽  
Vol 48 (4) ◽  
pp. 458-474
Author(s):  
Junaid Khan
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Standard Basis ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Grobner Basis

A theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial orderings which are not necessarily well orderings [1]) for ideals in polynomial rings over a field is developed. We call these bases “SASBI Basis” for “Subalgebra Analogue to Standard Basis for Ideals”. The case of global orderings, here they are called “SAGBI Basis” for “Subalgebra Analogue to Gröbner Basis for Ideals”, is treated in [6]. Sasbi bases may be infinite. In this paper we consider subalgebras admitting a finite Sasbi basis and give algorithms to compute them.

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 An extension of Buchberger’s criteria for Gröbner basis decision

2010 ◽  
Vol 13 ◽  
pp. 111-129
Author(s):  
John Perry
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Characterization Theorem ◽  
Polynomial Systems ◽  
Polynomial Ideal ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Case Scenario ◽  
Worst Case ◽  
New Criterion

AbstractTwo fundamental questions in the theory of Gröbner bases are decision (‘Is a basisGof a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known thatGis a Gröbner basis if and only if a certain set of polynomials (theS-polynomials) satisfy a certain property. In general there arem(m−1)/2 of these, wheremis the number of polynomials inG, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systemsGfor which the new criterion has dramatic impact, reducing the worst-case scenario fromm(m−1)/2 S-polynomials tom−1.

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〉.

scholarly journals A note on multivariate polynomial division and Gröbner bases

2015 ◽  
Vol 97 (111) ◽  
pp. 43-48
Author(s):  
Aleksandar Lipkovski ◽  
Samira Zeada
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Heuristic Approach ◽  
Grobner Bases ◽  
Multivariate Polynomials ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Leading Term ◽  
Monomial Ordering

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Grobner, and to suggest a new heuristic approach to Grobner basis computations.

Determining of Level Sets for a Fuzzy Surface Using Gröbner Basis

2015 ◽  
Vol 4 (2) ◽  
pp. 1-14
Author(s):  
Hamed Farahani ◽  
Sajjad Rahmany ◽  
Abdolali Basiri
Keyword(s):  
Level Sets ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Statistical Nature ◽  
Numerical Examples ◽  
Incomplete Datasets ◽  
The Given

In this paper, a manner to determine the level sets of a fuzzy surface using the benefits of Gröbner basis is presented. Fuzzy surfaces are constructed from incomplete datasets or from data that contain uncertainty which has not statistical nature. The authors firstly define the concept of level sets for the fuzzy surfaces. Then, employing Gröbner bases benefits a criterion is proposed for when the level sets of the fuzzy surface are nonempty sets. Moreover, a new algorithm is designed to determine the level sets. The big advantage of the proposed method lies in the fact that it attains all members of the level sets of the fuzzy surface at a time. Finally, some applied numerical examples are illustrated to demonstrate the proficiency of the given approach.

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