A counterexample to a conjecture of Bass, Connell and Wright

Piotr Ossowski

Colloquium Mathematicae (1998)

  • Volume: 77, Issue: 2, page 315-320
  • ISSN: 0010-1354

Abstract

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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=(G1,...,Gn) 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 ) = ∑ 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).

How to cite

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Ossowski, Piotr. "A counterexample to a conjecture of Bass, Connell and Wright." Colloquium Mathematicae 77.2 (1998): 315-320. <http://eudml.org/doc/210593>.

@article{Ossowski1998,
abstract = {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=(G1,...,Gn) 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).},
author = {Ossowski, Piotr},
journal = {Colloquium Mathematicae},
keywords = {polynomial automorphism; rooted trees; Jacobian conjecture},
language = {eng},
number = {2},
pages = {315-320},
title = {A counterexample to a conjecture of Bass, Connell and Wright},
url = {http://eudml.org/doc/210593},
volume = {77},
year = {1998},
}

TY - JOUR
AU - Ossowski, Piotr
TI - A counterexample to a conjecture of Bass, Connell and Wright
JO - Colloquium Mathematicae
PY - 1998
VL - 77
IS - 2
SP - 315
EP - 320
AB - 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=(G1,...,Gn) 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).
LA - eng
KW - polynomial automorphism; rooted trees; Jacobian conjecture
UR - http://eudml.org/doc/210593
ER -

References

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  1. [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. Zbl0539.13012
  2. [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. Zbl0831.13004
  3. [3] D. Wright, Formal inverse expansion and the Jacobian conjecture, J. Pure Appl. Algebra 48 (1987), 199-219. Zbl0666.12017
  4. [4] A. V. Yagzhev, On Keller's problem, Sibirsk. Mat. Zh. 21 (1980), no. 5, 141-150 (in Russian). 

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