Gaps in Polynomial Endomorphisms and the Jacobian Conjecture

Abstract

Let k be a field of characteristic zero. We say that a polynomial endomorphism F: k ⁿ  → k ⁿ has a gap after component m if the homogeneous components F _(j) are equal to 0 for j = m + 1,…, n(m + 1) +1. We prove the equivalence of the Jacobian conjecture and the following condition: if polynomial automorphism has a gap after component m, then the sum forms an automorphism.

Key Words:

• Gap
• Jacobian Conjecture
• Polynomial automorphisms

2000 Mathematics Subject Classification:

• Primary 14R15
• Secondary 14R10

Notes

Communicated by J.-T. Yu.