论文标题
古典群体的差异galo理论中的正常形式
Normal Forms in Differential Galois Theory for the Classical Groups
论文作者
论文摘要
令$ g $为一个经典的尺寸$ d $，然后让$ \ boldsymbol {a} =（a_1，\ dots，a_d）$是差分不确定的，这是零的特征零字段$ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ c $ c $。再让$ a（\ boldsymbol {a}）$为lie代数$ \ mathfrak {g}（f \ langle \ langle \ boldsymbol {a} \ rangle）$ g $的$ g $的$ g $的$ g $。众所周知（参见Juan的工作），$ \ boldsymbol {y}'= a（\ boldsymbol {a}）\ boldsymbol {y} $ a $ f \ f \ langle \ boldsymbol \ boldsymbol {a}} \ rangle $ g（c）$ g（c）$。在本文中，我们构建了一个差异字段扩展名$ \ natercal {l} $ of $ f \ langle \ boldsymbol {a} \ rangle $，以至于$ \ nathcal {l} $的常数领域是$ c $， $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $\mathcal{L}$ is still the full group $G(C)$ and $A(\boldsymbol{a})$ is gauge equivalent over $\mathcal{L}$ to a matrix in normal form which we introduced in work by Seiss.我们还考虑了$ a（\ boldsymbol {a}）$的系数的专业。
Let $G$ be a classical group of dimension $d$ and let $\boldsymbol{a}=(a_1,\dots,a_d)$ be differential indeterminates over a differential field $F$ of characteristic zero with algebraically closed field of constants $C$. Further let $A(\boldsymbol{a})$ be a generic element in the Lie algebra $\mathfrak{g}(F\langle \boldsymbol{a} \rangle)$ of $G$ obtained from parametrizing a basis of $\mathfrak{g}$ with the indeterminates $\boldsymbol{a}$. It is known (cf. work by Juan) that the differential Galois group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $F\langle \boldsymbol{a} \rangle$ is $G(C)$. In this paper we construct a differential field extension $\mathcal{L}$ of $F\langle \boldsymbol{a} \rangle$ such that the field of constants of $\mathcal{L}$ is $C$, the differential Galois group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $\mathcal{L}$ is still the full group $G(C)$ and $A(\boldsymbol{a})$ is gauge equivalent over $\mathcal{L}$ to a matrix in normal form which we introduced in work by Seiss. We also consider specializations of the coefficients of $A(\boldsymbol{a})$.
