On Generic differential  <IMG WIDTH="43" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="/proc/2008-136-04/S0002-9939-07-09314-8/gif-title0/img1.gif" ALT="$ \operatorname{SO}_n$">-extensions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On Generic differential $\operatorname{SO}_n$ -extensions

Authors:	Lourdes Juan Arne Ledet

Institution:	Department of Mathematics, Texas Tech University, MS 1042, Lubbock, Texas 79409 ; Department of Mathematics, Texas Tech University, MS 1042, Lubbock, Texas 79409

Abstract:	Let $\mathcal C$ be an algebraically closed field with trivial derivation and let $\mathcal F$ denote the differential rational field $\mathcal C\langle Y_{ij}\rangle$ , with $Y_{ij}$ , $1\leq i\leq n-1$ , $1\leq j\leq n$ , $i\leq j$ , differentially independent indeterminates over $\mathcal C$ . We show that there is a Picard-Vessiot extension $\mathcal E\supset \mathcal F$ for a matrix equation $X'=X\mathcal A(Y_{ij})$ , with differential Galois group $\operatorname{SO}_n$ , with the property that if is any differential field with field of constants $\mathcal C$ , then there is a Picard-Vessiot extension $E\supset F$ with differential Galois group $H\leq\operatorname{SO}_n$ if and only if there are $f_{ij}\in F$ with $\mathcal A(f_{ij})$ well defined and the equation $X'=X\mathcal A(f_{ij})$ giving rise to the extension $E\supset F$ .

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