# Generic properties of Jacobians of smooth functions

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  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.

 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.

 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.

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  reference, so the perturbations are by polynomials of certain degree; there is however some non-trivial linear algebra to deal with.