A counterexample to a conjecture of Bass, Connell and Wright

Ossowski, Piotr

Ossowski, Piotr

Colloquium Mathematicae, Tome 78 (1998), p. 315-320 / Harvested from The Polish Digital Mathematics Library

Résumé

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).

Publié le : 1998-01-01

Zbl 0942.13011

EUDML-ID : urn:eudml:doc:210593

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     author = {Piotr Ossowski},
     title = {A counterexample to a conjecture of Bass, Connell and Wright},
     journal = {Colloquium Mathematicae},
     volume = {78},
     year = {1998},
     pages = {315-320},
     zbl = {0942.13011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv77z2p315bwm}
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Ossowski, Piotr. A counterexample to a conjecture of Bass, Connell and Wright. Colloquium Mathematicae, Tome 78 (1998) pp. 315-320. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv77z2p315bwm/

Bibliographie

[000] [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. | Zbl 0539.13012

[001] [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. | Zbl 0831.13004

[002] [3] D. Wright, Formal inverse expansion and the Jacobian conjecture, J. Pure Appl. Algebra 48 (1987), 199-219. | Zbl 0666.12017

[003] [4] A. V. Yagzhev, On Keller's problem, Sibirsk. Mat. Zh. 21 (1980), no. 5, 141-150 (in Russian).