# A counterexample to a conjecture of Bass, Connell and Wright

Colloquium Mathematicae (1998)

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

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topOssowski, 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

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

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