Multiplicatively separable function

Definition

For a function of two variables

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

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

on the entire domain of $G$ .

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

$G (x_{1}, x_{2}, \dots, x_{n}) = g_{1} (x_{1}) g_{2} (x_{2}) \dots g_{n} (x_{n})$

(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 $A, B$ , $G$ is multiplicatively separable with respect to the partition if there exist functions $g_{A}, g_{B}$ such that:

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