Let $${M}_{m,n}$$ M m , n denote the space of $$m\times n$$ m × n matrices with entries in the formal power series ring $$K[[x_1,\ldots , x_s]], K$$ K [ [ x 1 , … , x s ] ] , K an arbitrary field. We consider different groups G acting on $$M_{m,n}$$ M m , n by formal change of coordinates, combined with the multiplication by invertible matrices. This includes right and contact equivalence of functions, mappings, and ideals. A matrix A is called finitely G -determined if any matrix B , with entries of $$A-B$$ A - B in $$\langle x_1,\ldots ,x_s\rangle ^k$$ ⟨ x 1 , … , x s ⟩ k for some k , is contained in the G -orbit of A . In this paper we present algorithms to check finite determinacy, to compute determinacy bounds and to compute the tangent image $${\widetilde{T}}_A(GA)$$ T ~ A ( G A ) of the action. The tangent image is an important invariant in positive characteristic since it differs in general from the tangent space $$T_A(GA)$$ T A ( G A ) to the orbit of G (in a subtle way). This fact was only recently discovered by the authors and is proved in the present paper by using our algorithms. Besides this application, the algorithms of this paper are of interest for the classification of singularities in arbitrary characteristic, a subject of growing interest.

中文翻译：

---

任意特性群作用的算法与奇点理论中的一个问题

令 $${M}_{m,n}$$M m , n 表示 $$m\times n$$ m × n 个矩阵的空间，其中条目位于形式幂级数环 $$K[[x_1,\ ldots , x_s]], K$$ K [ [ x 1 , … , xs ] ] , K 任意字段。我们考虑通过坐标的形式变化，结合可逆矩阵的乘法作用于 $$M_{m,n}$$M m , n 的不同群 G。这包括函数、映射和理想的权利和接触等价。矩阵 A 被称为有限 G -确定如果任何矩阵 B ，在 $$\langle x_1,\ldots ,x_s\rangle ^k$$ ⟨ x 1 , … , xs ⟩ 中的条目 $$AB$$ A - B k 对于某些 k ，包含在 A 的 G 轨道中。在本文中，我们提出了检查有限确定性、计算确定性界限和计算动作的切线图像 $${\widetilde{T}}_A(GA)$$ T ~ A ( GA ) 的算法。切线图像是正特性的重要不变量，因为它与切线空间 $$T_A(GA)$$TA ( GA ) 到 G 的轨道大体不同（以微妙的方式）。这个事实最近才被作者发现，并在本文中通过使用我们的算法得到了证明。除了这个应用之外，本文的算法对于任意特征的奇点分类也很有意义，这是一个越来越受关注的主题。