A degree bound for strongly nilpotent polynomial automorphisms

Samuel G. G. Johnston

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)259-271
Number of pages13
JournalJournal of Algebra
Volume608
Early online date1 Jun 2022
DOIs
Publication statusPublished - 15 Oct 2022

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

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