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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 (fromto)  259271 
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
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 1 Finished

Random fragmentationcoalescence processes out of equilibrium
Kyprianou, A. & Rogers, T.
Engineering and Physical Sciences Research Council
30/03/20 → 31/12/22
Project: Research council