Order and dimension

It will be shown that the conjectured Jacobi bound for the order of differential systems cannot be valid if a very natural conjecture concerning the differential dimension of such systems is false. 1. Summary. It will be shown that the conjectured Jacobi bound for the order of differential systems cannot be valid if a very natural conjecture concerning the differential dimension of such systems is false. It is indeed obvious that in certain cases in which the Jacobi bound is -oo (so that it is considered to hold only if the system has no component of finite order) the Jacobi bound conjecture must be false if the conjecture concerning dimension is false. It will be shown that if the conjecture concerning dimension is false, then there are even cases in which the Jacobi bound is finite but invalid. 2. Notation. K is throughout an ordinary differential field of characteristic 0. Superscripts denote differentiation. 3. Introduction. Let 5 be a system of k differential polynomials in n > k differential indeterminates with coefficients in K. A very natural conjecture [8, Appendix, paragraph 10] is that if the manifold of S is not empty, then every component of the manifold is of positive differential dimension. This conjecture will be referred to as the dimension conjecture. If the dimension conjecture is false, then the strong Jacobi bound for order (definition below) must be invalid. For let S be a system which violates the dimension conjecture, and let T be obtained from S by adjoining the 0 polynomial n — k times. Then T is a system of « differential polynomials in n indeterminates. The strong Jacobi bound for T is -oo, but T has a component of differential dimension 0. The relation between the dimension conjecture and the conjecture that the strong Jacobi bound is valid when it is -oo has been investigated in detail by Joseph Tomasovic [9, §6] for partial as well as ordinary differential equations. He proves the equivalence of these conjectures as well as other results. We are left with the question of the relation between the dimension conjecture and the strong Jacobi bound in the case that this bound is finite. It is the purpose of this