Abstract
AbstractIt is shown that a polynomial map $$F: {\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$$
F
:
R
n
→
R
n
with nowhere zero Jacobian determinant is invertible if and only if an explicit auxiliary polynomial system admits only the trivial solution. The main corollary is a concrete invertibility criterion in the Jacobian conjecture. The proof, conceptually related to differential geometry, represents a simple but infrequent application of differential equations to algebra.
Publisher
Springer Science and Business Media LLC
Reference12 articles.
1. Bialynicki-Birula, A., Rosenlicht, M.: Injective morphisms of real algebraic varieties. Proc. Am. Math. Soc. 13, 200–203 (1962)
2. Serre, J.-P.: How to use finite fields for problemed concerning infinite fields. arXiv:0903.0517 [math.AG]
3. Pinchuk, S.: A counterexample to the strong real Jacobian conjecture. Math. Z. 217, 1–4 (1994)
4. Braun, F., Fernandes, F.: A counterexample to a folk real Jacobian conjecture (preprint)
5. Fernandes, F.: A new class of non-injective polynomial local diffeomorphisms on the plane. J. Math. Anal. Appl. 507, 125736 (2022)