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Multilinear nilalgebras and the Jacobian theorem

Dmitri Piontkovski

Abstract

If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then its Jacobian matrix is invertible. This refines a theorem of Gerstenhaber and partially answers a question posed by Dotsenko.

Multilinear nilalgebras and the Jacobian theorem

Abstract

If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then its Jacobian matrix is invertible. This refines a theorem of Gerstenhaber and partially answers a question posed by Dotsenko.

Paper Structure

This paper contains 3 theorems, 8 equations.

Key Result

Theorem 1

Each Gerstenhaber nil algebra is Engel.

Theorems & Definitions (5)

  • Theorem 1: Gerstenhaber, Umirbaev
  • Conjecture 2: Jacobian conjecture for homogeneous mapping
  • Conjecture 3: A Jacobian conjecture in Yagzhev form
  • Proposition 4: Jacobian theorem
  • Theorem 5: Jacobian theorem for an infinite set of variables