Power Linear Keller Maps of Nilpotency Index Two

Abstract

Let K be a field. A polynomial map K ⁿ  → K ⁿ is power linear of degree d if it is of the form X + H = (X ₁ + H ₁,…, X _n  + H _n ), where is a linear form in X ₁,…, X _n and d > 1. In this paper it is proved that if K is a field of characteristic not dividing d and F is a power linear polynomial map of degree d with nilpotency index two, i.e., (JH)² = 0, then there exists a linear invertible polynomial map Φ such that Φ⁻¹ FΦ is a triangular power linear map of degree d.

Key Words:

• Polynomial map
• Invertible map
• Linearly triangularizable map
• Tame
• Jacobian conjecture

Mathematics Subject Classification:

• Primary 14E09

Acknowledgments

Notes

^#Communicated by I. Swanson.