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IMAGES OF HIGHER-ORDER DIFFERENTIAL OPERATORS OF POLYNOMIAL ALGEBRAS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 96 / Issue 2 / October 2017
- Published online by Cambridge University Press:
- 09 June 2017, pp. 205-211
- Print publication:
- October 2017
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- Article
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We investigate images of higher-order differential operators of polynomial algebras over a field
$k$. We show that, when 
$\operatorname{char}k>0$, the image of the set of differential operators 
$\{\unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D70F}_{i}\mid i=1,2,\ldots ,n\}$ of the polynomial algebra 
$k[\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{n},z_{1},\ldots ,z_{n}]$ is a Mathieu subspace, where 
$\unicode[STIX]{x1D70F}_{i}\in k[\unicode[STIX]{x2202}_{z_{1}},\ldots ,\unicode[STIX]{x2202}_{z_{n}}]$ for 
$i=1,2,\ldots ,n$. We also show that, when 
$\operatorname{char}k=0$, the same conclusion holds for 
$n=1$. The problem concerning images of differential operators arose from the study of the Jacobian conjecture.
