Den Essen And
نویسنده
چکیده
Let z = (z1, · · · , zn) and ∆ = ∑n i=1 ∂ 2 ∂z i the Laplace operator. A formal power series P (z) is said to be Hessian Nilpotent(HN) if its Hessian matrix HesP (z) = ( ∂ 2 P ∂zi∂zj ) is nilpotent. In recent developments in [BE1], [M] and [Z], the Jacobian conjecture has been reduced to the following so-called vanishing conjecture(VC) of HN polynomials: for any homogeneous HN polynomial P (z) (of degree d = 4), we have ∆P(z) = 0 for any m >> 0. In this paper, we first show that, the VC holds for any homogeneous HN polynomial P (z) provided that the projective subvarieties ZP and Zσ2 of CP n−1 determined by the principal ideals generated by P (z) and σ2(z) := ∑n i=1 z i , respectively, intersect only at regular points of ZP . Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F = z−∇P with P (z) HN if F has no non-zero fixed point w ∈ C with ∑n i=1 w i = 0. Secondly, we show that the VC holds for a HN formal power series P (z) if and only if, for any polynomial f(z), ∆(f(z)P (z)) = 0 when m >> 0.
منابع مشابه
Der Bereich Information, Kommunikation und Medien (IKM) in der Universität Duisburg-Essen
Zusammenfassung: Am 1.1.2003 wurde die Universität Duisburg-Essen neu gegründet. Sie ist aus den Universitäten Duisburg und Essen hervorgegangen. Der sich daran anschließende Fusionsprozess ist komplex und vielfach schmerzlich gewesen. Er ist auch bei Weitem noch nicht abgeschlossen. In einer Fusion liegt immer auch eine Chance für einen Neuanfang, der Hochschulen in dieser Form normaler Weise ...
متن کاملAn Extension of the Miyanishi-sugie Cancellation Theorem to Dedekind Rings
The version of the following full text has not yet been defined or was untraceable and may differ from the publisher's version.
متن کاملA polynomial counterexample to the Markus-Yamabe Conjecture
We give a polynomial counterexample to both the Markus-Yamabe Conjecture and the discrete Markus-Yamabe problem for all dimensions ≥ 3.
متن کاملAn Algorithm concerning One Dimensional Rings of Constants in Polynomial Rings
The version of the following full text has not yet been defined or was untraceable and may differ from the publisher's version.
متن کامل