A few remarks on the Generalized Vanishing Conjecture

@article{Bondt2013AFR,
  title={A few remarks on the Generalized Vanishing Conjecture},
  author={Michiel de Bondt},
  journal={Archiv der Mathematik},
  year={2013},
  volume={100},
  pages={533-538}
}
  • M. D. Bondt
  • Published 13 June 2012
  • Mathematics
  • Archiv der Mathematik
AbstractWe show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lambda^m f^m = 0] \Longrightarrow \forall_{m \gg 0}[\Lambda^m (g f^m) = 0]$$for a fixed differential operator $${\Lambda \in k[\partial]}$$ follows from a special case of it, namely that the additional factor g is a power of the radical polynomial f. Next we show that in order to prove the Generalized Vanishing Conjecture (up to some bound on the degree of Λ), we may assume that Λ is a linear combination of powers… 
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References

SHOWING 1-10 OF 11 REFERENCES
A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS
In the recent progress (4), (17) and (25), the well-known JC (Ja- cobian conjecture) ((2), (10)) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent)
Hessian nilpotent polynomials and the Jacobian conjecture
Let z = (z 1 ,···,z n ) and let A = Σ n i=1 ∂ 2 ∂z 2 i be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is
The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
Introduction I. The Jacobian Conjecture 1. Statement of the Jacobian Problem; first observations 2. Some history of the Jacobian Conjecture 3. Faulty proofs 4. The use of stabilization and of formal
A reduction of the Jacobian conjecture to the symmetric case
The main result of this paper asserts that it suffices to prove the Jacobian Conjecture for all polynomial maps of the form x + H, where H is homogeneous (of degree 3) and JH is nilpotent and
Polynomial Automorphisms and the Jacobian Conjecture
In this paper we give an update survey of the most important results concerning the Jacobian conjecture: several equivalent descriptions are given and various related conjectures are discussed. At
A primer of algebraic D-modules
1. The Weyl algebra 2. Ideal structure of the Weyl algebra 3. Rings of differential operators 4. Jacobian conjectures 5. Modules over the Weyl algebra 6. Differential equations 7. Graded and filtered
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