# American Institute of Mathematical Sciences

January  2002, 8(1): 17-28. doi: 10.3934/dcds.2002.8.17

## Global inversion via the Palais-Smale condition

 1 Department of Mathematics, Texas Christian University, Fort Worth, TX 76129, United States 2 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States

Received  March 2001 Revised  October 2001 Published  October 2001

Fixing a complete Riemannian metric g on $\mathbb R^n$, we show that a local diffeomorphism $f : \mathbb R^n\to \mathbb R^n$ is bijective if the height function $f\cdot v$ (standard inner product) satisfies the Palais-Smale condition relative to $g$ for each for each nonzero $v\in \mathbb R^n$. Our method substantially improves a global inverse function theorem of Hadamard. In the context of polynomial maps, we obtain new criteria for invertibility in terms of Lojasiewicz exponents and tameness of polynomials.
Citation: Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17
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