## Abstract

Let k be a field of characteristic zero. Let F=X+H be a polynomial map from k
^{n} to k
^{n}, where X is the identity map and H has only degree two terms and higher. We say that the Jacobian matrix JH of H is strongly nilpotent with index p if for all X
^{(1)},…,X
^{(p)}∈k
^{n} we have JH(X
^{(1)})…JH(X
^{(p)})=0. Every F of this form is a polynomial automorphism, i.e. there is a second polynomial map F
^{−1} such that F∘F
^{−1}=F
^{−1}∘F=X. We prove that the degree of the inverse F
^{−1} satisfies deg(F
^{−1})≤deg(F)
^{p−1}, improving in the strongly nilpotent case on the well known degree bound deg(F
^{−1})≤deg(F)
^{n−1} for general polynomial automorphisms.

Original language | English |
---|---|

Pages (from-to) | 259-271 |

Number of pages | 13 |

Journal | Journal of Algebra |

Volume | 608 |

Early online date | 1 Jun 2022 |

DOIs | |

Publication status | Published - 15 Oct 2022 |

## Keywords

- Formal inversion
- Jacobian conjecture
- Polynomial automorphism
- Strongly nilpotent

## ASJC Scopus subject areas

- Algebra and Number Theory