Jacobian-squared function-germs

In this paper, it is shown that, for any equidimensional $C^{\mathrm{\infty }}$ map-germ $f:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$, the map-germ $F: ($\mathbb{R}$^n, 0) \to $\mathbb{R}$^n\times$\mathbb{R}$^{\ell}$definedby$F(x)=\left(f(x), \mu_1(x){|Jf|^2(x)}, \cdots, \mu_\ell(x){|Jf|^2(x)}\right)$ is always a frontal; where ${\mu }_i$ is a $C^{\mathrm{\infty }}$ function-germ and $\mathrm{\mid }Jf\mathrm{\mid }$ is the Jacobian-determinant of $f$. Moreover, it is also shown that when the multiplicity of $f$ is less than or equal to $3$, any frontal constructed from $f$ must be $\mathcal{A}$-equivalent to a frontal $F$ of the above form.