Re-embeddings of Affine Algebras Via Gröbner Fans of Linear Ideals
Martin Kreuzer, Le Ngoc Long, Lorenzo Robbiano
TL;DR
We address the problem of re-embedding affine $K$-algebras $R=P/I$ with fewer indeterminates by replacing elimination-based Gröbner computations with the Gröbner fan of the linear part $\mathrm{Lin}_{\mathfrak{M}}(I)$, where $\mathfrak{M}=\langle x_1,\dots,x_n\rangle$. The core advance is showing that suitable candidate separating tuples $Z$ come from $\mathrm{GFan}(\mathrm{Lin}_{\mathfrak{M}}(I))$, and providing efficient, linear-algebraic methods to compute $\mathrm{GFan}(I_L)$ for $I_L=\langle \mathrm{Lin}_{\mathfrak{M}}(I)\rangle$ via maximal minors of the coefficient matrix, plus a cotangent-equivalence framework for binomial linear ideals to simplify classification. These results yield an algorithmic pipeline to obtain (optimal) $Z$-separating re-embeddings and to certify optimality, with practical applications to border basis schemes where the linear parts are binomial. The paper also includes implementation details and demonstrates that border basis moduli can often be realized as affine spaces after re-embedding, enabling significant dimension reduction in computations and moduli problems.
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 $R=P'/I'$ 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 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.