We determine a Gröbner basis for the secant ideal of the toric ideal associated to the second hypersimplex ∆(2, n), with respect to any circular term order. The Gröbner basis of the secant ideal requires polynomials of odd degree up to n. This shows that the circular term order is 2-delightful, resolving a conjecture of Drton, Sturmfels, and the author. The proof uses Gröbner degenerations for ...

In this paper we describe an efficient involutive algorithm for constructing Gröbner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain way. In the presented algorithm a reduced Gröbner basis is the internally fixed subset of an involutive basis, and having computed the later, the former can be...

We show that the set of cluster monomials for the cluster algebra of type D4 contains a basis of the Z-module Z[x1,1, . . . , x3,3]. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases....

The GVW algorithm, presented by Gao et al., is a signature-based algorithm for computing Gröbner bases. In this paper, a variant of GVW is presented. This new algorithm is called a monomial-oriented GVW algorithm or mo-GVW algorithm for short. The moGVW algorithm presents a new frame of GVW and regards labeled monomials instead of labeled polynomials as basic elements of the algorithm. Being di...

In this paper we examine subalgebras on two generators in the univariate polynomial ring. A set, S, of polynomials in a subalgebra of a polynomial ring is called a canonical basis (also referred to as SAGBI basis) for the subalgebra if all lead monomials in the subalgebra are products of lead monomials of polynomials in S. In this paper we prove that a pair of polynomials {f, g} is a canonical ...

We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Gröbner bases, factorization or sub-resultant computations.

(2) Let n be an ordinal number, T be an admissible connected term order of n, L be an addassociative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and f , p, g be polynomials of n, L. Suppose f reduces to g, p, T . Then there exists a monomial m of n, L such that g = f −m∗ p and HT(m∗ p,T ) ...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

We construct a monomial basis of the positive part U of the quantized enveloping algebra associated to a finite–dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of U. 0. Introduction In [L1], Lusztig showed that the positive part U of the quantized enveloping algebra associated to a finite–dimensional simple Lie alge...

Classical methods to study and solve systems of polynomial equations are based on numerous avatars of Gröbner (standard) basis algorithms or Riquier-Janet type methods (Ritt-Wu’s algorithm). All these methods use implicitly but deeply the dense or sparse representation of multivariate polynomials, which is the computer science counterpart of the expansion of these mathematical objects on the mo...