Re-embeddings of Affine Algebras Via Gröbner Fans of Linear Ideals

Abstract

Given an affine algebra $R=K[x_1,\dots ,x_n]/I$ over a field $K$, where $I$ is an ideal in the polynomial ring $P=K[x_1,\dots ,x_n]$, we examine the task of effectively calculating re-embeddings of $I$, i.e., of presentations $R=P^{\prime }/I^{\prime }$ such that $P^{\prime }=K[y_1,\dots ,y_m]$ has fewer indeterminates. For cases when the number of indeterminates $n$ is large and Gröbner basis computations are infeasible, we have previously introduced the method of $Z$-separating re-embeddings. This method tries to detect polynomials of a special shape in $I$ which allow us to eliminate the indeterminates in the tuple $Z$ by a simple substitution process. Here we improve this approach by showing that suitable candidate tuples $Z$ can be found using the Gröbner fan of the linear part of $I$. Then we describe a method to compute the Gröbner fan of a linear ideal, and we improve this computation in the case of binomial linear ideals using a cotangent equivalence relation. Finally, we apply the improved technique in the case of the defining ideals of border basis schemes.

arXiv comment21 pages. Minor corrections and improvements. To appear in "Beiträge zur Algebra und Geometrie". arXiv admin note: substantial text overlap with arXiv:2207.08115