# 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 [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.

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.