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Colloquium Mathematicum

1998 | 77 | 2 | 315-320
Tytuł artykułu

A counterexample to a conjecture of Bass, Connell and Wright

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let F=X-H:$k^n$ → $k^n$ be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G_1,...,G_n) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of $G_i$ of degree 2d+1 can be expressed as $G_i^{(d)}=\sum_T α(T)^{-1} σ_i(T)$, where T varies over rooted trees with d vertices, α(T)=CardAut(T) and $σ_i(T)$ is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, $F$ is an automorphism or, equivalently, $G_i^{(d)}$ is zero for sufficiently large d. Bass, Connell and Wright conjecture that not only $G_i^{(d)}$ but also the polynomials $σ_i(T)$ are zero for large d. The aim of the paper is to show that for the polynomial automorphism (4) and rooted trees (3), the polynomial $σ_2(T_s)$ is non-zero for any index $s$ (Proposition 4), yielding a counterexample to the above conjecture (see Theorem 5).
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
315-320
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-10-29
poprawiono
1998-01-09
Twórcy
autor
• Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Bibliografia
• [1] H. Bass, E. H. Connell and D. Wright, The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 287-330.
• [2] A. van den Essen, A counterexample to a conjecture of Meisters, in: Automorphisms of Affine Spaces, Proc. Internat. Conf. on Invertible Polynomial Maps (Curaçao, 1994), Kluwer, 1995, 231-233.
• [3] D. Wright, Formal inverse expansion and the Jacobian conjecture, J. Pure Appl. Algebra 48 (1987), 199-219.
• [4] A. V. Yagzhev, On Keller's problem, Sibirsk. Mat. Zh. 21 (1980), no. 5, 141-150 (in Russian).
Typ dokumentu
Bibliografia
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