The Jacobian Conjecture in case of "non-negative coefficients"

• Autorzy: Ludwik M. Drużkowski
• Języki publikacji: EN

Abstrakty

EN

It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form
$F(x₁,...,x_n) = x - H(x) := (x₁ - H₁(x₁,...,x_n),...,x_n - H_n(x₁,...,x_n))$,
where $H_j$ are homogeneous polynomials of degree 3 with real coefficients (or $H_j = 0$), j = 1,...,n and H'(x) is a nilpotent matrix for each $x = (x₁,...,x_n) ∈ ℝ^n$.
We give another proof of Yu's theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case
$deg F^{-1} ≤ (deg F)^{ind F - 1}$, where $ind F := max{ind H'(x): x ∈ ℝ^n}$.
Note that the above inequality is not true when the coefficients of H are arbitrary real numbers; cf. [E3].

Słowa kluczowe

EN

polynomial automorphisms; nilpotent matrix; Jacobian Conjecture

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1997
• Tom: 66
• Numer: 1
• Strony: 67-75

Daty

wydano

1997

otrzymano

1995-07-30

Twórcy

autor

Ludwik M. Drużkowski

• Institute of Mathematics, Jagiellonian University, Reymonta 4/508, 30-059 Kraków, Poland

Bibliografia

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