Tutorial on Differential Galois Theory II

TutorialonDifferential Galois Theory IIT. DyckerhoffDepartment of MathematicsUniversity of Pennsylvania02/13/08 / OberflockenbachOutlineToday’s planPicard Vessiot ringsThe ∂ Galois group schemeThe Torsor theorem and applicationsDescent theory for Picard Vessiot extensionsn th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ... .. .    . . ..n−1 n−1∂ (y) −a −a −a ... −a ∂ (y)0 1 2 n−1⇒ We develop Picard Vessiot theory for general systems of1 st order equations:n×n∂(y) = Ay with A∈ Fwhich we denote by [A].Systems of ∂ equationsYesterday we considered:a field F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 i⇒ We develop Picard Vessiot theory for general systems of1 st order equations:n×n∂(y) = Ay with A∈ Fwhich we denote by [A].Systems of ∂ equationsYesterday we considered:a field F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 in th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ... . . .    . . ..n−1 n−1∂ (y) −a −a −a ... −a ∂ (y)0 1 2 n−1Systems of ∂ equationsYesterday we considered:a field F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 in th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ...