Let $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$ be analytic functions that passes through origin, i.e., $f(\boldsymbol 0) = g(\boldsymbol 0) = 0$. Is true that if the product $fg$ is a monomial,i.e., has simple normal crossings, then $f$ and $g$ have simple normal crossings? I am asking this because if we have a desingularization for the zero set for the product, then each function has simple normal crossings, which is already quite good for me.
This result was taken from Lemma 3.4. The worst part is that he does not have proven the result, and I am not sure if @lulu counterexemple fits as a counterexemple. Here what is needed is just a desingularized set of functions $\{f,g\}$ in the author's definition, which is what I have asked.