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Symmetric Jacobians

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This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over ℂ such that satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d ≥ 3.
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Bibliografia
  • [1] de Bondt M., Quasi-translations and counterexamples to the homogeneous dependence problem, Proc. Amer. Math. Soc., 2006, 134(10), 2849–2856 http://dx.doi.org/10.1090/S0002-9939-06-08335-3
  • [2] de Bondt M.C., Homogeneous Keller Maps, PhD thesis, Radboud University Nijmegen, 2009, available at http://webdoc.ubn.ru.nl/mono/b/bondt_m_de/homokema.pdf
  • [3] de Bondt M., Constant polynomial Hessian determinants in dimension three, preprint available at http://arxiv.org/abs/1203.6605
  • [4] de Bondt M., van den Essen A., Singular Hessians, J. Algebra, 2004, 282(1), 195–204 http://dx.doi.org/10.1016/j.jalgebra.2004.08.026
  • [5] de Bondt M., van den Essen A., A reduction of the Jacobian Conjecture to the symmetric case, Proc. Amer. Math. Soc., 2005, 133(8), 2201–2205 http://dx.doi.org/10.1090/S0002-9939-05-07570-2
  • [6] Dillen F., Polynomials with constant Hessian determinant, J. Pure Appl. Algebra, 1991, 71(1), 13–18 http://dx.doi.org/10.1016/0022-4049(91)90037-3
  • [7] Druzkowski L.M., New reduction in the Jacobian conjecture, Univ. Iagel. Acta Math., 2001, 39, 203–206
  • [8] Druzkowski L.M., The Jacobian conjecture: symmetric reduction and solution in the symmetric cubic linear case, Ann. Polon. Math., 2005, 87, 83–92 http://dx.doi.org/10.4064/ap87-0-7
  • [9] van den Essen A., Polynomial Automorphisms and the Jacobian Conjecture, Progr. Math., 190, Birkhäuser, Basel, 2000
  • [10] van den Essen A.R.P., Hubbers E., A new class of invertible polynomial maps, J. Algebra, 1997, 187(1), 214–226 http://dx.doi.org/10.1006/jabr.1997.6788
  • [11] Gordan P., Nöther M., Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet, Math. Ann., 1876, 10(4), 547–568 http://dx.doi.org/10.1007/BF01442264
  • [12] Meng G., Legendre transform, Hessian conjecture and tree formula, Appl. Math. Lett., 2006, 19(6), 503–510 http://dx.doi.org/10.1016/j.aml.2005.07.006
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bwmeta1.element.doi-10_2478_s11533-013-0393-7
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