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On product decomposition

Given a finite set $W$ in $\bar{k}^n$ where $\bar{k}$ is the algebraic closure of a field $k$ one would like to determine if $W$ can be decomposed as $\prod_{i=1}^n V_i$ where $V_i \subset \bar{k}$ under a linear transformation, that is, $W\stackrelλ{\to} \prod_{i=1}^n V_i$ where $λ\in Gl_n (\bar{k})$. We assume that $W$ is presented as $W=Z(\mathcal{F})$, the zero set of a polynomial system $\mathcal{F}$ in $n$ variables over $k$. We study algebraic characterization of such product decomposition. For decomposition into component sets of the same cardinality we obtain a stronger characterization and show that the decomposition in this case is essentially unique (up to permutation and scalar multiplication of coordinates). We investigate computational problems that arise from the decomposition problem.