(A somewhat technical question, but maybe it is well known.)

Consider matrices over the ring $k[[x_1,\dots,x_n]]$, whose entries vanish at the origin (i.e. belong to the maximal ideal $\mathfrak{m}$). Denote by $J(A_{k,l})$ the ideal of maximal minors of the matrix $A_{k,l}\in Mat(k,l,\mathfrak{m})$.

Given two generic enough (and mutually generic!) matrices, consider the ideal generated by all the maximal minors: $\Big( J(A_{k_1,l_1}),J(B_{k_2,l_2}) \Big)$. (Assume $k_1\le l_1$ and $k_2\le l_2$.)

'{\bf Q.}': When does this ideal contain $\mathfrak{m}^{k_1+k_2-1}$?

Of course, an obvious necessary condition is: $(l_1-k_1+1)+(l_2-k_2+1)\ge n$, but is this condition sufficient?

example 1. Suppose $k_1=l_1$, $k_2=l_2$, then for the generic case we have: det(A) is of order $k_1$ and $det(B)$ is of order $k_2$ and the lowest order parts of the two polynomials form a regular sequence. Thus, if $n\le 2$ we get: $(det(A),det(B))\supset\mathfrak{m}^{k_1+k_2-1}$.

example 2. Suppose $k_1=1$, take $A_{1,l}$ with generic entries of first order. Again, one can show that the needed property holds.

What about the general case? How to address such questions?