Systems of $n$ partial differential equations in $n$ unknown functions: the conjecture of M. Janet

Abstract:

It was conjectured by Janet that an analytic solution to a system of n “independent” analytic differential equations in n unknown functions if not isolated must depend on at least one unknown function of $m - 1$ variables plus possibly other functions of fewer than m variables. Here m is the dimension of the complex domain on which the equations and the solution are given. An algebraic generalization of the linear form of the conjecture is proven. Also the result is extended to give a nonlinear version.