Systems of n Partial Differential Equations in n Unknown Functions: the Conjecture of M. Janet

Joseph Johnson
1978 Transactions of the American Mathematical Society  
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
more » ... sion. The purpose of this note is to show that the conjecture of Maurice Janet stated in [1] is a corollary of a result by Goodearl (Theorem 7 of [2] ). That is done in Theorem 1. In Theorem 2, a nonlinear generalization of Janet's conjecture is proved. The reader who would like to know in advance the original form of Janet's conjecture should read the corollary to Theorem 2 and the discussion that follows the corollary.
doi:10.2307/1997742 fatcat:xefgqt3qtra3negszehpw7awzm