In 1939 Keller conjectured that any polynomial mapping $f\colon\C^n\to\C^n$ with constant nonvanishing Jacobian determinant, should be invertible. This open problem bears the name of Jacobian conjecture. Druzkowski proved that cubic-linear mappings are sufficient to decide the conjecture. For this important class we develop an algorithm that translates the constant Jacobian condition into algebraic equations in the matrix of parameters. We also single out a natural special case of these conditions, that we call D-nilpotency. The class of D-nilpotent matrices turns out to coincide with set of matrices that are permutation-similar to upper-triangular matrices. The corresponding cubic-linear maps are always invertible.

Druzkowski matrix search and D-nilpotent automorphisms

ZAMPIERI, Gaetano
1999-01-01

Abstract

In 1939 Keller conjectured that any polynomial mapping $f\colon\C^n\to\C^n$ with constant nonvanishing Jacobian determinant, should be invertible. This open problem bears the name of Jacobian conjecture. Druzkowski proved that cubic-linear mappings are sufficient to decide the conjecture. For this important class we develop an algorithm that translates the constant Jacobian condition into algebraic equations in the matrix of parameters. We also single out a natural special case of these conditions, that we call D-nilpotency. The class of D-nilpotent matrices turns out to coincide with set of matrices that are permutation-similar to upper-triangular matrices. The corresponding cubic-linear maps are always invertible.
1999
Jacobian conjecture; cubic-linear functions; D-nilpotent matrices
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/236602
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 5
  • ???jsp.display-item.citation.isi??? 6
social impact