Additive & Multiplicative Link

Question

Is there an isolative property about the following multivariable equation: $$f_1(x)g_1(y)=f_2(x)+g_2(y)$$ That is, is this equation rearrangable such that it may be put in the form $\Omega_1(x)=\Omega_2(y)$ where $\Omega_1(x)=\Omega_1(f_1(x),f_2(x))$ & $\Omega_2(x)=\Omega_2(g_1(x),g_2(x))$

Answer 1

It seems the following.

To make the arithmetic operations defined, suppose that $\operatorname{ran} f_i, \operatorname{ran} g_i\subset\Bbb R$. Also we shall interpret the equality as $\operatorname{dom} f_1=\operatorname{dom} f_2=D_f$ and $\operatorname{dom} g_1=\operatorname{dom} g_2=D_g$. To avoid trivialities, we shall assume that $D_f, D_g\ne\varnothing$. Fix points $x_0\in D_f$, $y_0\in D_g$. Then

$$f_2(x)=f_1(x)g_1(y_0)-g_2(y_0),$$

$$g_2(y)=f_1(x_0)g_1(y)-f_2(x_0).$$

Hence $$(f_1(x)-f_1(x_0)) (g_1(y)-g_1(y_0))=f_1(x_0)g_1(y_0)-f_2(x_0)-g_2(y_0)=0.$$

So either the function $f_1$ is constant (and, hence, the function $f_2$ is constant too) or the function $g_1$ is constant (and, hence, the function $g_2$ is constant too).