transactions of the american mathematical society Volume 242, August 1978 SYSTEMS OF n PARTIAL DIFFERENTIAL EQUATIONS IN n UNKNOWN FUNCTIONS: THE CONJECTURE OF M. JANET BY JOSEPH JOHNSON 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. 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. 1. GoodearFs result. If A is a finite set and R a differential ring (possibly noncommutative) with A as its fundamental set of (commuting) derivation operators we will call R a A-ring. In fact generally we will write "A" where formerly we would have written "differential." Let us write tyR for the ring of linear A-operators with coefficients in R. We will think of the elements of tyü as polynomials in A with coefficients in R written on the left. Naturally in öD*, if 1 < i,j < m and a E R, 5,-5y = 0,5,- and 8¡a = S,(a) + aS¡. For the reader's convenience, the following paraphrase of Goodearl's result is given here. In its statement pd stands for "projective dimension." Theorem (Goodearl). Let the A-ring R be semiprime and right and left noetherian. If M is a nonzero right tf)R-module and finitely generated as a right R-module then pdq, M = card A + pd^M. Let m E N and fix a set A with m elements 8X,..., 8m. Let k be a Received by the editors April 20, 1977. AMS (MOS) subject classifications (1970).Primary 12H05. Key words and phrases. Differential algebra, differential ring, differential module, Kahler differentials. © American Mathematical Society 1978 329