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

Annales Polonici Mathematici (1997)

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

## Access Full Article

top## Abstract

top## How to cite

topLudwik 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

top- [BCW] H. Bass, E. H. Connell and D. Wright, The Jacobian Conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982), 287-330. Zbl0539.13012
- [BR] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203. Zbl0107.14602
- [D1] L. M. Drużkowski, An effective approach to Keller's Jacobian Conjecture, Math. Ann. 264 (1983), 303-313. Zbl0504.13006
- [D2] L. M. Drużkowski, The Jacobian Conjecture in case of rank or corank less than three, J. Pure Appl. Algebra 85 (1993), 233-244. Zbl0781.13015
- [D3] L. M. Drużkowski, The Jacobian Conjecture, preprint 492, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1991.
- [D4] L. M. Drużkowski, The Jacobian Conjecture: some steps towards solution, in: Automorphisms of Affine Spaces, A. van den Essen (ed.), Kluwer, 1995, 41-53. Zbl0839.13012
- [DR] L. M. Drużkowski and K. Rusek, The formal inverse and the Jacobian Conjecture, Ann. Polon. Math. 46 (1985), 85-90. Zbl0644.12010
- [E1] A. van den Essen, Polynomial maps and the Jacobian Conjecture, Report 9034, Catholic University, Nijmegen, 1990.
- [E2] A. van den Essen, The exotic world of invertible polynomial maps, Nieuw Arch. Wisk. (4) 11 (1) (1993), 21-31. Zbl0802.13003
- [E3] A. van den Essen, A counterexample to a conjecture of Drużkowski and Rusek, Ann. Polon. Math. 62 (1995), 173-176. Zbl0838.14008
- [K] O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), 299-306.
- [KS] T. Krasiński and S. Spodzieja, On linear differential operators related to the n-dimensional Jacobian Conjecture, in: Real Algebraic Geometry, M. Coste, L. Mahé and M.-F. Roy (eds.), Lecture Notes in Math. 1524, Springer, 1992, 308-315. Zbl0774.32008
- [M] G. H. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mountain J. Math. 12 (1982), 679-705. Zbl0523.34050
- [MO] G. H. Meisters and C. Olech, A poly-flow formulation of the Jacobian Conjecture, Bull. Polish Acad. Sci. Math. 35 (1987), 725-731. Zbl0646.34018
- [P] S. Pinchuk, A counterexample to the real Jacobian Conjecture, Math. Z. 217 (1994), 1-4. Zbl0874.26008
- [R] K. Rusek, Polynomial automorphisms, preprint 456, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1989.
- [RW] K. Rusek and T. Winiarski, Polynomial automorphisms of ${C}^{n}$, Univ. Iagel. Acta Math. 24 (1984), 143-149.
- [S] Y. Stein, The Jacobian problem as a system of ordinary differential equations, Israel J. Math. 89 (1995), 301-319. Zbl0823.34010
- [W] T. Winiarski, Inverse of polynomial automorphisms of ${C}^{n}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 673-674. Zbl0451.32014
- [Y] A. V. Yagzhev, On Keller's problem, Sibirsk. Mat. Zh. 21 (1980), 141-150 (in Russian). Zbl0466.13009
- [Yu] J.-T. Yu, On the Jacobian Conjecture: reduction of coefficients, J. Algebra 171 (1995), 515-523. Zbl0816.13017

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.