相关论文: Groebner-Shirshov Basis for the Chinese Monoid
We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products Grothendieck polynomials in each set of…
Garside calculus is the common mechanism that underlies a certain type of normal form for the elements of a monoid, a group, or a category. Originating from Garside's approach to Artin's braid groups, it has been extended to more and more…
We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the…
A general formula is presented for any order derivative of Chebyshev polynomials instead of the existing recursive relationship. Hence, the Chebyshev finite difference method is made applicable not only to second order problems but also to…
Polytopes in R^n with integral vertices form a monoid under the Minkowski sum, and the Grothendieck construction gives rise to a group. We show that every symmetric polytope is a norm in this group for every n.
In a recent paper Lima, Panario and Wang have provided a new method to multiply polynomials in Chebyshev basis which aims at reducing the total number of multiplication when polynomials have small degree. Their idea is to use Karatsuba's…
We establish the Gr\"obner-Shirshov bases theory for differential Lie $\Omega$-algebras. As an application, we give a linear basis of a free differential Lie Rota-Baxter algebra on a set.
In this paper, we give decision criteria for normal binomial difference polynomial ideals in the univariate difference polynomial ring F{y} to have finite difference Groebner bases and an algorithm to compute the finite difference Groebner…
We introduce a Bernstein-Bezier basis for the pyramid, whose restriction to the face reduces to the Bernstein-Bezier basis on the triangle or quadrilateral. The basis satisfies the standard positivity and partition of unity properties…
We use the rational tableaux introduced by Stembridge to give a bideterminant basis for a normal reductive monoid and for its variety of noninvertible elements. We also obtain a bideterminant basis for the full coordinate ring of the…
We provide a polynomial time algorithm for computing the universal Gr\"obner basis of any polynomial ideal having a finite set of common zeros in fixed number of variables. One ingredient of our algorithm is an effective construction of the…
We consider the natural monoid structure on the set of quadratic rings over an arbitrary base scheme and characterize this monoid in terms of discriminants.
In this paper, we characterized the relationship between Groebner bases and u-bases: any minimal Groebner basis of the syzygy module for n univariate polynomials with respect to the term-over-position monomial order is its u-basis.…
This article provides a method for constructing invariants and semi-invariants of a binary $N$-ic form over a field $k$ characteristics $0$ or $p > N$. A practical and broadly applicable sufficient condition for ensuring nontriviality of…
We propose a new algorithm solving the extended gcd problem, which provides a solution minimizing one of the two coordinates. The algorithm relies on elementary arithmetic properties.
In this paper, the tropical differential Gr\"obner basis is studied, which is a natural generalization of the tropical Gr\"obner basis to the recently introduced tropical differential algebra. Like the differential Gr\"obner basis, the…
We propose a novel encoding scheme for algebraic codes such as codes on algebraic curves, multidimensional cyclic codes, and hyperbolic cascaded Reed-Solomon codes and present numerical examples. We employ the recurrence from the Gr\"obner…
The Householder's method is a root-find algorithm which is a natural extension of the methods of Newton and Halley. The current paper mostly focuses on approximating the square root of a positive real number based on these methods. The…
We prove new characterizations based on Gr\"obner bases for the Cohen-Macaulay property of a projective monomial curve.
We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The…