Root Parametrized Differential Equations for the classical groups

Abstract

Let $C⟨t_1,\dots t_l⟩$ be the differential field generated by $l$ differential indeterminates $\mathbf{t}=(t_1,\dots ,t_l)$ over an algebraically closed field $C$ of characteristic zero. We develop a lower bound criterion for the differential Galois group $G(C)$ of a matrix parameter differential equation $\partial (\mathbf{y})=A(\mathbf{t})\mathbf{y}$ over $C⟨t_1,\dots t_l⟩$ and we prove that every connected linear algebraic group is the Galois group of a linear parameter differential equation over $C⟨t_1⟩$. As a second application we compute explicit and nice linear parameter differential equations over $C⟨t_1,\dots ,t_l⟩$ for the groups ${\mathrm{SL}}_{l+1}(C)$, ${\mathrm{SP}}_{2l}(C)$, ${\mathrm{SO}}_{2l+1}(C)$, ${\mathrm{SO}}_{2l}(C)$, i.e. for the classical groups of type $A_l$, $B_l$, $C_l$, $D_l$, and for ${\mathrm{G}}_2$ (here $l=2$).

New version with corrections September 2020

References in corpus (3)

• The inverse problem of differential Galois theory over the field R(z)
• Picard-Vessiot theory of differentially simple rings
• A Root Parametrized Differential Equation for the Special Linear Group

Cited by in corpus (1)

• On General Extension Fields for the Classical Groups in Differential Galois Theory