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A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)

A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)

Let \(k\) be a field of characteristic zero. For any polynomial mapping \(F=(F_1,\ldots,F_n):k^n\rightarrow k^n\) by multidegree of \(F\) we mean the following \(n\)-tuple of natural numbers mdeg \(F=(\deg F_1,\ldots,\deg F_n).\) Let us denote by \(k[x]=k[x_1,\ldots,x_n]\) a ring of polynomials in \...

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Main Authors: Karaś, M., Pękała, P.
Format: Article
Language:English
Published: Lugansk National Taras Shevchenko University 2023
Subjects:
derivation
locally nilpotent derivation
polynomial automorphism
multidegree
13N15; 14R10; 16W20
Online Access:https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2042
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spelling oai:ojs.admjournal.luguniv.edu.ua:article-20422023-12-11T16:21:07Z A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\) Karaś, M. Pękała, P. derivation, locally nilpotent derivation, polynomial automorphism, multidegree 13N15; 14R10; 16W20 Let \(k\) be a field of characteristic zero. For any polynomial mapping \(F=(F_1,\ldots,F_n):k^n\rightarrow k^n\) by multidegree of \(F\) we mean the following \(n\)-tuple of natural numbers mdeg \(F=(\deg F_1,\ldots,\deg F_n).\) Let us denote by \(k[x]=k[x_1,\ldots,x_n]\) a ring of polynomials in \(n\) variables \(x_1,\ldots,x_n\) over \(k.\) If \(D:k[x]\rightarrow k[x]\) is a locally nilpotent \(k\)-derivation, then one can define the automorphism \(\exp D\) of \(k\)-algebra \(k[x]\) and then the polynomial automorphism \((\exp D)_{\star}\) of \(k^n\). In this note we present a general upper bound of mdeg \((\exp D)_{\star}\) in the case of a triangular derivation \(D\), and also show that this estimataion is exact. Lugansk National Taras Shevchenko University 2023-12-11 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2042 10.12958/adm2042 Algebra and Discrete Mathematics; Vol 36, No 1 (2023) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2042/pdf Copyright (c) 2023 Algebra and Discrete Mathematics
institution Algebra and Discrete Mathematics
collection OJS
language English
topic derivation
locally nilpotent derivation
polynomial automorphism
multidegree
13N15; 14R10; 16W20
spellingShingle derivation
locally nilpotent derivation
polynomial automorphism
multidegree
13N15; 14R10; 16W20
Karaś, M.
Pękała, P.
A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)
topic_facet derivation
locally nilpotent derivation
polynomial automorphism
multidegree
13N15; 14R10; 16W20
format Article
author Karaś, M.
Pękała, P.
author_facet Karaś, M.
Pękała, P.
author_sort Karaś, M.
title A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)
title_short A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)
title_full A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)
title_fullStr A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)
title_full_unstemmed A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)
title_sort note on multidegrees of automorphisms of the form \((\exp d)_{\star}\)
description Let \(k\) be a field of characteristic zero. For any polynomial mapping \(F=(F_1,\ldots,F_n):k^n\rightarrow k^n\) by multidegree of \(F\) we mean the following \(n\)-tuple of natural numbers mdeg \(F=(\deg F_1,\ldots,\deg F_n).\) Let us denote by \(k[x]=k[x_1,\ldots,x_n]\) a ring of polynomials in \(n\) variables \(x_1,\ldots,x_n\) over \(k.\) If \(D:k[x]\rightarrow k[x]\) is a locally nilpotent \(k\)-derivation, then one can define the automorphism \(\exp D\) of \(k\)-algebra \(k[x]\) and then the polynomial automorphism \((\exp D)_{\star}\) of \(k^n\). In this note we present a general upper bound of mdeg \((\exp D)_{\star}\) in the case of a triangular derivation \(D\), and also show that this estimataion is exact.
publisher Lugansk National Taras Shevchenko University
publishDate 2023
url https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2042
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first_indexed 2024-04-12T06:25:41Z
last_indexed 2024-04-12T06:25:41Z
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