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 \(R=P'/I'\) such that \(P'=K[y_1,\dots ,y_m]\) has fewer indeterminates. For cases when the number of indeterminates n is large and Gröbner basis computations are infeasible, we have introduced the method of Z-separating re-embeddings in Kreuzer et al. (J Algebra Appl 21, 2022) and Kreuzer, et al. (São Paulo J Math Sci, 2022). 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.
