What $n$-linear sums can be extracted from a product of linear polynomials in $m$ variables?

Question

Let $\boldsymbol{c}_1, ..., \boldsymbol{c}_n$ be $n$ orthonormal, $m$-dimensional complex vectors, with $\boldsymbol{c}_i = (c_{i,1}, ..., c_{i,m})$. Consider the following polynomial in $x_1,..., x_m$: $$ (c_{11} x_1 + c_{12} x_2 + ... + c_{1m} x_m) (c_{21} x_1 + c_{22} x_2 + ... + c_{2m} x_m) \cdots (c_{n1} x_1 + c_{n2} x_2 + ... + c_{nm} x_m) = \prod_{i=1}^n \sum_{j=1}^m c_{ij} x_j. \tag{A} $$ Is there a way to control the resulting sums of terms containing all different $x_i$? With this I mean, given a target normalised complex vector $\boldsymbol v$ of length $\binom{m}{n}$, can I find a set of orthogonal $\boldsymbol{c}_i$ such that the $n$-linear terms of (A), renormalised, give $\boldsymbol v$?

As an example, in the $m=3, n=2$ case, (A) can be written as $$ (c_{11} c_{21} x_1^2 + c_{12} c_{22} x_2^2 + c_{13} c_{23} x_3^2) + (c_{11} c_{22} + c_{12} c_{21}) x_1 x_2 + (c_{11} c_{23} + c_{13} c_{21}) x_1 x_3 + (c_{12} c_{23} + c_{13} c_{22}) x_2 x_3, $$ so that the problem amounts to solving the following nonlinear system: \begin{align} c_{11} c_{22} + c_{12} c_{21} &= N v_1, \\ c_{11} c_{23} + c_{13} c_{21} &= N v_2, \\ c_{12} c_{23} + c_{13} c_{22} &= N v_3, \end{align} for some normalisation constant $N$ and $\boldsymbol c_1, \boldsymbol c_2$ constrained to be orthonormal.

In the general case this is clearly related to the problem of computing permanents of submatrices of a matrix $C$ having the $\boldsymbol c_i$ as row/columns.

Has this kind of thing been studied? Is there relevant literature I can look at?

Answer 1

Yes something close to your question has been studied before see these MO questions:

1. Grassmann-Plücker relations for permanents
2. which homogeneous polynomials split into linear factors?

Basically there is a rather complicated system of algebraic equations of degree $n+1$ called the Brill-Gordan equations for the coefficients of your polynomial which tell you if it is a product of linear factors. You would need more work to eliminate the coefficients of the non-multilinear terms in order to get a system of equation for the multilinear terms only. I don't know what the effect of imposing orthogonality would be.