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

Given an affine algebra $R=K[x_1,\dots ,x_n]\mathrm{/}I$ 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 $R=P^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}$ such that $P^{\mathrm{\prime }}=K[y_1,\dots ,y_m]$ 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.