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Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 915–933 | Cite as

On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers

  • Tomáš Bajbar
  • Oliver Stein
Article

Abstract

In this article we analyze the global diffeomorphism property of polynomial maps \(F:\mathbb {R}^n\rightarrow \mathbb {R}^n\) by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials \(\Vert F\Vert _2^2\). This allows us to identify a class of polynomial maps F for which their global diffeomorphism property on \(\mathbb {R}^n\) is equivalent to their Jacobian determinant \(\det JF\) vanishing nowhere on \(\mathbb {R}^n\). In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.

Keywords

Newton polytope Coercivity Global invertibility Real Jacobian Conjecture Circuit number 

Mathematics Subject Classification

14P99 26B10 26C05 52B20 

Notes

Acknowledgements

The authors are grateful to Yu. Nesterov and V. Shikhman for pointing out the importance of the invariance of coercivity under linear transformations, and for other fruitful discussions on the subject of this article. The authors also wish to thank an anonymous referee for substantial remarks which significantly improved the quality of this manuscript.

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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Institute for MathematicsGoethe-UniversityFrankfurt am MainGermany
  2. 2.Institute of Operations ResearchKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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