Let $Y_1, Y_2,\dots, Y_n$ be $n$ bases of linear forms in the polynomial ring $k[x_1,\dots, x_n]$, where $k$ is a field or $\mathbb{Z}$. Examples of bases $Y_i$'s are $A_i\cdot[x_1,\dots, x_n]^T$, where $A_i\in \operatorname{GL}_n(k)$. For each $1\leq i\leq n$, let $e_i$ denote the usual elementary symmetric polynomial in $n$ variables. When is the sequence $e_1(Y_1), e_2(Y_2), \dots, e_n(Y_n)$ a regular sequence in $k[x_1,\dots, x_n]$? Here $e_i(Y_i)$ denotes the $i^{th}$ elementary symmetric polynomial in the variables $Y_i$. Obviously this is true if all the $Y_i$'s are equal. I am mainly interested in the case when $Y_i$'s are not all equal.
