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

Ludwik M. Drużkowski

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 67-75
  • ISSN: 0066-2216

Abstract

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It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form F ( x ₁ , . . . , x n ) = x - H ( x ) : = ( x ₁ - H ₁ ( x ₁ , . . . , x n ) , . . . , x n - H n ( x ₁ , . . . , 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 = ( x ₁ , . . . , 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 d e g F - 1 ≤ ( d e g F ) i n d F - 1 , where i n d F : = m a x i n d H ' ( x ) : x ∈ ℝ n . Note that the above inequality is not true when the coefficients of H are arbitrary real numbers; cf. [E3].

How to cite

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Ludwik M. Drużkowski. "The Jacobian Conjecture in case of "non-negative coefficients"." Annales Polonici Mathematici 66.1 (1997): 67-75. <http://eudml.org/doc/269937>.

@article{LudwikM1997,
abstract = {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 $ind F := max\{ind H^\{\prime \}(x): x ∈ ℝ^n\}$. Note that the above inequality is not true when the coefficients of H are arbitrary real numbers; cf. [E3].},
author = {Ludwik M. Drużkowski},
journal = {Annales Polonici Mathematici},
keywords = {polynomial automorphisms; nilpotent matrix; Jacobian Conjecture; real jacobian conjecture},
language = {eng},
number = {1},
pages = {67-75},
title = {The Jacobian Conjecture in case of "non-negative coefficients"},
url = {http://eudml.org/doc/269937},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Ludwik M. Drużkowski
TI - The Jacobian Conjecture in case of "non-negative coefficients"
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 67
EP - 75
AB - 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 $ind F := max{ind H^{\prime }(x): x ∈ ℝ^n}$. Note that the above inequality is not true when the coefficients of H are arbitrary real numbers; cf. [E3].
LA - eng
KW - polynomial automorphisms; nilpotent matrix; Jacobian Conjecture; real jacobian conjecture
UR - http://eudml.org/doc/269937
ER -

References

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