The inverse Gröbner basis problem is to find the ideals that have a given monomial ideal as its initial ideal. We consider the problem of finding when the given monomial ideal is the initial ideal of a prime ideal. Kalkbrenner and Sturmfels (1995), in Theorem 1, prove that the radical of the initial ideal of a prime ideal is equi-dimensional and connected in codimension one or equivalently, the...

We show that the toric ideal of a lattice path polymatroid is generated by quadrics corresponding to symmetric exchanges, and give a monomial order under which these quadrics form a Gröbner basis. We then obtain an analogous result for lattice path matroids.

A basis of Lorentz and gauge-invariant monomials in non-Abelian gauge theories with matter is described, applicable for the inverse mass expansion of effective actions. An algorithm to convert a arbitrarily given invariant expression into a linear combination of the basis elements is presented. The linear independence of the basis invariants is proven.

A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems t...

Let IF be a field, V ⊆ IF be a (combinatorially interesting) finite set of points. Several important properties of V are reflected by the polynomial functions on V . To study these, one often considers I(V ), the vanishing ideal of V in the polynomial ring IF[x1, . . . , xn]. Gröbner bases and standard monomials of I(V ) appear to be useful in this context, leading to structural results on V . ...

We study the relationship between certain Gröbner bases for zerodimensional radical ideals, and the varieties defined by the ideals. Such a variety is a finite set of points in an affine n-dimensional space. We are interested in monomial orders that “eliminate” one variable, say z. Eliminating z corresponds to projecting points in n-space to (n − 1)-space by discarding the z-coordinate. We show...

In this paper we consider a free associative algebra on three generators over an arbitrary field K. Given a term ordering on the commutative polynomial ring on three variables over K, we construct uncountably many liftings of this term ordering to a monomial ordering on the free associative algebra. These monomial orderings are total well orderings on the set of monomials, resulting in a set of...

It gives a class of p-Borel principal ideals of a polynomial algebra over a field K for which the graded Betti numbers do not depend on the characteristic of K and the Koszul homology modules have monomial cyclic basis. Also it shows that all principal p-Borel ideals have binomial cycle basis on Koszul homology modules.