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1997 | 66 | 1 | 67-75
Tytuł artykułu

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

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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].
Rocznik
Tom
66
Numer
1
Strony
67-75
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-07-30
Twórcy
  • Institute of Mathematics, Jagiellonian University, Reymonta 4/508, 30-059 Kraków, Poland
Bibliografia
  • [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.
  • [BR] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203.
  • [D1] L. M. Drużkowski, An effective approach to Keller's Jacobian Conjecture, Math. Ann. 264 (1983), 303-313.
  • [D2] L. M. Drużkowski, The Jacobian Conjecture in case of rank or corank less than three, J. Pure Appl. Algebra 85 (1993), 233-244.
  • [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.
  • [DR] L. M. Drużkowski and K. Rusek, The formal inverse and the Jacobian Conjecture, Ann. Polon. Math. 46 (1985), 85-90.
  • [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.
  • [E3] A. van den Essen, A counterexample to a conjecture of Drużkowski and Rusek, Ann. Polon. Math. 62 (1995), 173-176.
  • [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.
  • [M] G. H. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mountain J. Math. 12 (1982), 679-705.
  • [MO] G. H. Meisters and C. Olech, A poly-flow formulation of the Jacobian Conjecture, Bull. Polish Acad. Sci. Math. 35 (1987), 725-731.
  • [P] S. Pinchuk, A counterexample to the real Jacobian Conjecture, Math. Z. 217 (1994), 1-4.
  • [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.
  • [W] T. Winiarski, Inverse of polynomial automorphisms of $C^n$, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 673-674.
  • [Y] A. V. Yagzhev, On Keller's problem, Sibirsk. Mat. Zh. 21 (1980), 141-150 (in Russian).
  • [Yu] J.-T. Yu, On the Jacobian Conjecture: reduction of coefficients, J. Algebra 171 (1995), 515-523.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-apmv66z1p67bwm
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