Jacobian-squared function-germs

TL;DR

This paper demonstrates that a specific class of map-germs derived from Jacobian-squared functions are always frontals, and characterizes all such frontals for low multiplicity cases as being equivalent to these forms.

Contribution

It introduces a new class of frontals constructed from Jacobian-squared functions and classifies all low-multiplicity frontals as equivalent to these forms.

Findings

All Jacobian-squared derived map-germs are frontals.

For multiplicity ≤ 3, all frontals are equivalent to the Jacobian-squared form.

The paper provides a classification of low-multiplicity frontals.

Abstract

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 $F(x)=\left(f(x), \mu_1(x){|Jf|^2(x)}, \cdots, \mu_\ell(x){|Jf|^2(x)}\right)$ is always a frontal; where $\mu_i$ is a $C^\infty$ function-germ and $|Jf|$ 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.