Picard-Vessiot theory of partial differential fields

Abstract

Introduction.' The purpose of this note is to extend to partial differential fields the theory of homogeneous linear ordinary differential equations.2 Although this can be done by extending the procedure of PV to partial differential fields, it is perhaps not without interest to see how the theory in the partial case can be deduced from the ordinary theory, and the latter is what is done below. If 7 is an ordinary differential field of characteristic 0 with algebraically closed field of constants and 9 is an of 7 with the same field of constants, then (PV, ?17) 9 is a of 7 if there exists a homogeneous linear differential equation y(n) +ply(n-1) + * +pny=0 (each PiGC) with a fundamental system of solutions 71, * * , 7n such that G=J(o,, , qn)4 Since pi is then plus or minus the quotient of two determinants Wi/W0, where Wl=det (7(j) )<j<n,j9n-1;<k<n, it is the same thing to require that 9 contain elements qj, , tn linearly independent over constants such that G=f(t0, , On) and such that each Wi/W0 belongs to 7. It is this property which is generalized below to define Picard-Vessiot extension in the case of partial differential fields.