Reduction numbers and initial ideals

Face Numbers and Nongeneric Initial Ideals

Certain necessary conditions on the face numbers and Betti numbers of simplicial complexes endowed with a proper action of a prime order cyclic group are established. A notion of colored algebraic shifting is defined and its properties are studied. As an application a new simple proof of the characterization of the flag face numbers of balanced Cohen-Macaulay complexes originally due to Stanley...

Deterministically Computing Reduction Numbers of Polynomial Ideals

We present two approaches to compute the (absolute) reduction number of a polynomial ideal. The first one puts the ideal into a position such that the reduction number of its leading ideal can be easily read off the minimal generators and then uses linear algebra to determine the reduction number of the ideal itself. The second method computes via a Gröbner system not only the absolute reductio...

Initial Ideals of Truncated Homogeneous Ideals

Denote by R the power series ring in countably many variables over a eld K; then R 0 is the smallest sub-algebra of R that contains all homogeneous elements. It is a fact that a homogeneous, nitely generated ideal J in R 0 have an initial ideal gr(J), with respect to an arbitrary admissible order, that is locally nitely generated in the sense that dimK gr(J) d P d?1 j=1 R 0 j gr(J) d?j < 1 for ...

Lifting Problem in Codimension 2 and Initial Ideals

Linear Components and the Behavior of Graded Betti Numbers under the Transition to Generic Initial Ideals

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. In this paper we study the graded Betti numbers of graded ideals in S and E. First, we prove that if the graded Betti number β ii+k(S/I) = β S ii+k(S/Gin(I)) for some i > 1 and k ≥ 0 then one has β qq+k(S/I) = β S qq+k(S/Gin(I)) for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show...