2023/06/14 by Kreuzer, Martin, Long, Le Ngoc, Robbiano, Lorenzo
#13E15 #13P10 (Secondary) #14Q20 (Primary) 14R10 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.08481
Given an affine algebra R=K[x1,…,xn]/I over 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 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.