Let $f = (f_1, \dotso, f_n):\mathbb{R}^n \to \mathbb{R}^n$ be a smooth map and let $J$ be its Jacobian (determinant of the matrix with $ij$-th entry $\partial_i f_j$). We introduce the zero sets of $J$ and its derivatives $$Z_0 = J^{-1}(0), \quad Z_1 = Z_0 \cap (\nabla J)^{-1}(0), \quad \dotso$$ Here we define $Z_k = Z_{k - 1} \cap (\nabla^k J)^{-1}(0)$ for $k \geq 1$; $\nabla^k$ denotes the operation of taking $k$-th order derivatives. My question is: what are the generic properties of $J$? Do we have $Z_1 = \emptyset$, $Z_2 = \emptyset$ or $Z_k = \emptyset$ for some $k$, generically, and what are the relations between $k$ and $n$? Do we have for all $n$, that $Z_k = \emptyset$ generically for some $k$? It is fine to work on a compact set and consider generic properties on it, so $f: K \subset \mathbb{R}^n \to \mathbb{R}^n$ for some compact $K$.

A certain property is generic, if it holds on an open and dense set of $C^\infty(\mathbb{R}^n; \mathbb{R}^n)$ of maps between $\mathbb{R}^n \to \mathbb{R}^n$.

For example, for $n = 2$, Whitney [1] showed that generically we have $Z_1 = \emptyset$. His idea was to perturb $f$ by polynomials in parameter families and prove that the relation on $Z_k$ is large enough codimension. For $n = 3$, I believe we have the same result (computation). For $n \geq 4$ the set $Z_1$ is too big and we need to include $Z_2$, but I don't know counterexamples to this; the calculations also become increasingly difficult. In [2, section 1], there are some results related to stability/instability of maps by Thom, which need sufficient dimension.

[1] Whitney, Hassler On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane. Ann. of Math. (2) 62 (1955), 374–410.

[2] Arnolʹd, V. I. Singularity theory. Selected papers. Translated from the Russian. With an introduction by C. T. C. Wall. London Mathematical Society Lecture Note Series, 53. Cambridge University Press, Cambridge-New York, 1981.


1 Answer 1


This is answered in the paper https://link.springer.com/content/pdf/10.1007/s00526-020-01740-6.pdf (OA), Lemma 6.13; this Lemma is proved in the Appendix of the paper. The final answer is that one needs to take $k(n) = \left \lceil{n + 1}\right \rceil $ derivatives to obtain that generically $Z_{k(n) - 1} = \emptyset$. The ideas used are the ones from the Whitney [1] reference, so the perturbations are by polynomials of certain degree; there is however some non-trivial linear algebra to deal with.


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