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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| Datum: | 2023 |
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| Format: | Artikel |
| Sprache: | English |
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Lugansk National Taras Shevchenko University
2023
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2042 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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admjournalluguniveduua-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 |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2023-12-11T16:21:07Z |
| 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 |
| work_keys_str_mv |
AT karasm anoteonmultidegreesofautomorphismsoftheformexpdstar AT pekałap anoteonmultidegreesofautomorphismsoftheformexpdstar AT karasm noteonmultidegreesofautomorphismsoftheformexpdstar AT pekałap noteonmultidegreesofautomorphismsoftheformexpdstar |
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2025-12-02T15:42:26Z |
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2025-12-02T15:42:26Z |
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