Jacobian-squared function-germs (Q1756166)

Let $f: (\mathbb R^n, 0) \rightarrow (\mathbb R^n, 0)$ be a $C^\infty$ map-germ and let $\delta(f)$ denote the Jacobian determinant of $f$. The author calls the square of this determinant the Jacobian-squared function-germ of $f$. His aim is to show that $\delta^2(f)$ plays an important role in the construction of non-trivial frontals associated with a given equidimensional map-germ $f$. For example, it is easy to prove that for any sequence $g_i: (\mathbb R^n, 0) \rightarrow (\mathbb R, 0)$, $0\leqslant i\leqslant k$, the map-germ $F: (\mathbb R^n, 0) \rightarrow (\mathbb R^{n+k}, 0)$, defined by the rule $F=(f,\delta^2(f)g_1,\dots, \delta^2(f)g_k)$, is always a frontal. Moreover, the author shows that if the multiplicity of $f$ is less than or equal to 3, any frontal associated with $f$ is $\mathcal A$-equivalent to $F$ of the above form. In case $n=2$ his results are illustrated by a series of examples from the lists in [\textit{J. H. Rieger}, J. Lond. Math. Soc., II. Ser. 36, No. 1--2, 351--369 (1987; Zbl 0639.58007)], involving fold singularities, cuspidal edges and crosscaps, swallowtails, and so on.