@article{KreuzerLongRobbiano2024, author = {Kreuzer, Martin and Long, Le Ngoc and Robbiano, Lorenzo}, title = {Re-embeddings of affine algebras via Gr{\"o}bner fans of linear ideals}, series = {Beitr{\"a}ge zur Algebra und Geometrie / Contributions to Algebra and Geometry}, volume = {65}, journal = {Beitr{\"a}ge zur Algebra und Geometrie / Contributions to Algebra and Geometry}, number = {4}, publisher = {Springer Berlin Heidelberg}, address = {Berlin/Heidelberg}, issn = {0138-4821}, doi = {10.1007/s13366-024-00733-2}, url = {http://nbn-resolving.de/urn:nbn:de:101:1-2024041316352253862922}, pages = {827 -- 851}, year = {2024}, abstract = {Given an affine algebra R=K[x1,⋯,xn]/Iover a field  K , where I is an ideal in the polynomial ring P=K[x1,⋯,xn], we examine the task of effectively calculating re-embeddings of  I , i.e., of presentations R=P′/I′such that P′=K[y1,⋯,ym]has fewer indeterminates. For cases when the number of indeterminates  n is large and Gr{\"o}bner 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. (S{\~a}o 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 Gr{\"o}bner fan of the linear part of  I . Then we describe a method to compute the Gr{\"o}bner 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.}, language = {en} }