Sets of zero-dimensional ideals in the polynomial ring $\mathbf{k}[x,y]$ that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Gröbner cells. Conca-Valla and Constantinescu parametrize such Gröbner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively.

In this paper, we give a parametrization of $(x,y)$-primary ideals in Gröbner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by defining a notion of canonical Hilbert-Burch matrices of zero-dimensional ideals in the power series ring $\mathbf{k}〚x,y〛$ with a given leading term ideal with respect to a local term ordering.

多项式环中的零维理想集 $\mathbf{ķ}[X，ÿ]$就给定的术语排序而言，具有相同的理想主术语的已知仿射空间称为Gröbner细胞。Conca-Valla和Constantinescu将某些Gröbner细胞参数化为字典法和程度字典法排序的某些规范Hilbert-Burch矩阵。

在本文中，我们给出了参数化 $（X，ÿ）$-Gröbner单元中的基本理想，它与此类理想的局部结构兼容。更准确地说，我们通过定义幂级数环中零维理想的规范希尔伯特-伯奇矩阵的概念，将先前的结果扩展到局部设置$\mathbf{ķ}〚X，ÿ〛$ 对于本地术语排序，具有给定的主导术语理想。