Jacobian-squared function-germs

In this paper, it is shown that, for any equidimensional $C^{\infty}$ map-germ $f : (\mathbb{R}^n , 0) \to (\mathbb{R}^n, 0)$, the map-germ $F : (\mathbb{R}^n, 0) \to \mathbb{R}^n \times \mathbb{R}^{\ell}$ defined by $F(x) = \left ( f(x), \mu_1(x) {\lvert Jf \lvert}^2 (x), \dotsc, \mu_{\ell} (x) {\lvert Jf \rvert}^2 (x) \right )$ is always a frontal, where $\mu_i$ is a $C^{\infty}$ function-germ and $\lvert Jf \rvert$ 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.

Keywords

Jacobian-squared, frontal, opening, ramification module, equidimensional map-germ

2010 Mathematics Subject Classification

57R45, 58K05

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This work was supported by JSPS KAKENHI Grant Number 17K05245.

Received 17 August 2018

Published 21 December 2018