3 problems

Let $A\in M_{r,s}$ be a matrix over the relevant formal power-series ring, and let $G$ act on $A$. Write $\widetilde{T}_A(GA)$ for the reduced tangent space to the orbit of $A$. Th…

Let $K$ be the ground field, let $R=K[[x_1,\ldots,x_s]]$ with maximal ideal $\mathfrak{m}$, and let $A\in\mathfrak{m}\cdot M_{m,n}$ be a matrix. Let $G$ act on $M_{m,n}$ by the equ…

Let $I=(f_1,\dots,f_k)\cdot\mathbb{K}\{x\}$ be a complete intersection ideal, and let $I_\mu$ denote the ideal generated by $\mu$-jets of its generators. Write $\mathfrak{N}(I)$ fo…