The Typical Dimension of a System of First-Order Differential Equations

We prove that if a system of first-order partial differential equations in one variable has a nonzero Kolchin dimension polynomial, then its leading coefficient is equal to 1. The notion of typical differential dimension plays an important role in differential algebra. Some of its estimations were proved by J. Ritt and E. Kolchin; they also advanced several conjectures that were later refuted. There are bounds for the typical differential dimension in codimension 1 (E. Kolchin) and in the case of linear differential equations (D. Grigoriev). Note that in codimension 2 for systems of linear differential equations in one indeterminate the well-known B´ezout theorem holds, and in the case of several variables, we have earlier proved its analogue, which is not satisfied in higher codimensions. For nonlinear systems, in the general case, there are no exponential bounds yet (although it is known that the growth of the typical dimension is bounded by the Ackermann function).