Characterization of the *-subalgebras in the algebra of bounded operators acting on Hilbert space is presented. Sufficient conditions for the existence of a faithful representation in pre-Hilbert space of a *-algebra in terms of its Groebner basis are given. These conditions are generalization of the unshrinkability of monomial *-algebras introduced by C. Lance and P. Tapper. The applications t...

Sparse elimination exploits the structure of a set of multivariate polynomials by measuring complexity in terms of Newton polytopes. We examine polynomial systems that generate 0-dimensional ideals: a generic monomial basis for the coordinate ring of such a system is de ned from a mixed subdivision. We o er a simple proof of this known fact and relate the computation of a monomial basis to the ...

In 1982 Richard P. Stanley conjectured that any finitely generated Zn-graded module M over a finitely generated Nn-graded K-algebra R can be decomposed in a direct sum M = ⊕t i=1 νi Si of finitely many free modules νi Si which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least...

We study a graded algebra D=D(L,G) over Z defined by a finite lattice L and a subset G in L, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice L, as the Chow ring of a smooth toric variety that we c...

We determine a term order on the monomials in the variables Xij , 1 ≤ i < j ≤ n, such that corresponding initial ideal of the ideal of Pfaffians of degree r of a generic n by n skew-symmetric matrix is the Stanley-Reisner ideal of a join of a simplicial sphere and a simplex. Moreover, we demonstrate that the Pfaffians of the 2r by 2r skew-symmetric submatrices form a Gröbner basis for the given...

The Gröbner fan of an ideal I ⊂ k[x1, . . . , xn], defined by Mora and Robbiano, is a complex of polyhedral cones in Rn. The maximal cones of the fan are in bijection with the distinct monomial initial ideals of I as the term order varies. If I is homogeneous the Gröbner fan is complete and is the normal fan of the state polytope of I. In general the Gröbner fan is not complete and therefore no...

We consider algebras over a field K defined by a presentation K〈x1, . . . , xn : R〉, where R consists of ( n 2 ) square-free relations of the form xixj = xkxl with every monomial xixj , i 6= j, appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand...

In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Gröbner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations....