×

Nilpotent Jacobians in dimension three. (English) Zbl 1085.14052

It is known that if \(F=x+H:k^{n}\rightarrow k^{n}\) (\(k\) - a field of characteristic zero) is a polynomial Keller’s mapping (i.e. \(\text{Jac}~F= \text{const}.\neq 0\)) then the Jacobian matrix \(JH\) of \(H\) is nilpotent. It is known that to solve the Jacobian conjecture it suffices to study the case of homogeneous \(H.\) This, in turn, is related to the homogeneous dependence problem (which is still open): if \(H\) is homogeneous with \(JH\) nilpotent and \(H(0)=0,\) then \(H_{1},\ldots ,H_{n}\) are linearly dependent. In the inhomogeneous case the answer to the above problem is negative due to a counterexample given by the second author. The paper under review studies the polynomial mappings \(H\) with \(JH\) nilpotent in dimension \(n=3.\) The authors completely classify such mappings of the form \(H=(u(x,y),v(x,y),h(u(x,y),v(x,y)))\) with \(JH\) nilpotent for which components are linear independent.

MSC:

14R15 Jacobian problem
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Bass, H.; Connell, E.; Wright, D., The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc., 7, 287-330 (1982) · Zbl 0539.13012
[2] M. de Bondt, Quasi-translations and counterexamples to the homogeneous dependence problem, J. Algebra (2005) to appear.; M. de Bondt, Quasi-translations and counterexamples to the homogeneous dependence problem, J. Algebra (2005) to appear. · Zbl 1107.14054
[3] M. de Bondt, A. van den Essen, The Jacobian conjecture: linear triangularization for homogeneous polynomial maps in dimension three, Report 0413, University of Nijmegen, The Netherlands, 2004.; M. de Bondt, A. van den Essen, The Jacobian conjecture: linear triangularization for homogeneous polynomial maps in dimension three, Report 0413, University of Nijmegen, The Netherlands, 2004. · Zbl 1085.14053
[4] Cima, A.; Gasull, A.; Mañosas, F., The discrete Markus-Yamabe problem, Nonlinear Anal.: Theory Methods Appl., 35, 3, 343-354 (1999) · Zbl 0919.34042
[5] A. van den Essen, Nilpotent Jacobian matrices with independent rows, Report 9603, University of Nijmegen, The Netherlands, 1996.; A. van den Essen, Nilpotent Jacobian matrices with independent rows, Report 9603, University of Nijmegen, The Netherlands, 1996.
[6] A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser, Basel, 2000.; A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser, Basel, 2000. · Zbl 0962.14037
[7] E. Hubbers, The Jacobian conjecture: cubic homogeneous maps in dimension four, Master’s Thesis, University of Nijmegen, The Netherlands, 1994.; E. Hubbers, The Jacobian conjecture: cubic homogeneous maps in dimension four, Master’s Thesis, University of Nijmegen, The Netherlands, 1994.
[8] Meisters, G., Polymorphisms conjugate to dilations, (Automorphisms of Affine Spaces, Proc. of the Conf. on Invertible Polynomial Maps held in Curacao. Automorphisms of Affine Spaces, Proc. of the Conf. on Invertible Polynomial Maps held in Curacao, July 4-8, 1994 (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 67-88 · Zbl 0856.14002
[9] Olech, C., On Markus-Yamabe stability conjecture, (Proc. of the Intern. Meeting on Ordinary Differential Equations and their Applications (1995), University of Florence), 127-137
[10] K. Rusek, Polynomial automorphisms, Preprint 456, Institute of Mathematics, Polish Academy of Sciences, IMPAN, Śniadeckich 8, PO Box 137, 00-950, Warsaw, Poland, 1989, preprint.; K. Rusek, Polynomial automorphisms, Preprint 456, Institute of Mathematics, Polish Academy of Sciences, IMPAN, Śniadeckich 8, PO Box 137, 00-950, Warsaw, Poland, 1989, preprint.
[11] Wright, D., The Jacobian conjecture: linear triangularization for cubics in dimension three, Linear and Multilinear Algebra, 34, 85-97 (1993) · Zbl 0811.13014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.