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 \...
Збережено в:
Видавець: | Lugansk National Taras Shevchenko University |
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Дата: | 2023 |
Автори: | , |
Формат: | Стаття |
Мова: | English |
Опубліковано: |
Lugansk National Taras Shevchenko University
2023
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Теми: | |
Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2042 |
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Організація
Algebra and Discrete MathematicsРезюме: | 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. |
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