Multiplicatively separable function

Definition

For a function of two variables

Suppose $F$ is a function of two variables $x$ and $y$. We say that $F$ is multiplicatively separable if there exist functions $f,g$ of one variable such that:

$F(x,y)=f(x)g(y)$

on the entire domain of $F$.

Note that the concept of multiplicatively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally multiplicatively separable need not remain multiplicatively separable.

For a function of many variables

Suppose $F$ is a function of $n$ variables $x_{1},x_{2},\dots ,x_{n}$. We say that $F$ is completely multiplicatively separable if there exist functions $f_{1},f_{2},\dots ,f_{n}$, each a function of one variable, such that:

$F(x_{1},x_{2},\dots ,x_{n})=f_{1}(x_{1})f_{2}(x_{2})\dots f_{n}(x_{n})$

(note that the subscripts here are not to be confused with subscripts used for partial derivatives).

There is a weaker notion of partially multiplicatively separable: if we express the set $\{1,2,\dots ,n\}$ as a union of two disjoint subsets $A,B$, $F$ is multiplicatively separable with respect to the partition if there exist functions $f_{A},f_{B}$ such that:

$F(x_{1},x_{2},\dots ,x_{n})=f_{A}({\mbox{only the variables }}x_{i},i\in A)f_{B}({\mbox{only the variables }}x_{i},i\in B)$