Can anyone find necessary and sufficient conditions on real-valued $f_1,f_2,$ and $f_3$ that allow us to equate coefficients in the equation delineated in the title?
I'm assuming linear independence will be needed to make the statement precise, but I'm not too sure how. Maybe we can place conditions by examining the Taylor expansion of these functions centered about $(0,0,0)$?
All I have now is that $f_1,f_2,$ and $f_3$ cannot contain any linear terms, but I'm not too sure if that's even true.
Thank you for all of your help, and be well.