Cotangent spaces and separating re-embeddings

Given an affine algebra [Formula: see text], where [Formula: see text] is a polynomial ring over a field [Formula: see text] and [Formula: see text] is an ideal in [Formula: see text], we study re-embeddings of the affine scheme [Formula: see text], i.e. presentations [Formula: see text] such that [Formula: see text] is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials [Formula: see text] in the ideal [Formula: see text] which are coherently separating in the sense that they are of the form [Formula: see text] with an indeterminate [Formula: see text] which divides neither a term in the support of [Formula: see text] nor in the support of [Formula: see text] for [Formula: see text]. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of [Formula: see text]. The dimension of the cotangent space of [Formula: see text] at a [Formula: see text]-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of [Formula: see text] and found an optimal re-embedding.

Small Gröbner fans of ideals of points

In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Gröbner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this property. Moreover, we provide methodologies for constructing such ideals. We then relax the condition of uniqueness. The second and most relevant topic discussed here is to consider and identify pairs of ideals with the same number of reduced Gröbner bases, that is, with the same cardinality of their associated Gröbner fan.