An existence theorem for systems of implicit differential equations

This note is based on the thesis [1] of the first author written under the guidance of the second author. The main technical input is Theorem 6 below. It will be proved in more generality in the subsequent paper [4]. Let f1, ..., fl be differential polynomials in one derivative and N variables with coefficients in IR. Suppose I ⊆ IR is an open interval and c : I −→ IR is a C∞-map with f1(c(t)) = ... = fl(c(t)) = 0 (t ∈ I). Let a be the differential ideal generated by f1, ..., fl in the differential polynomial ring IR{X1, ..., XN}. Then a is certainly a semi real ideal, i.e. for all g1, ..., gm ∈ IR{X1, ..., XN} we have 1 + ∑m j=1 g 2 j 6∈ a. This follows immediately from our assumption that c is a differential solution of the generators f1, ..., fl of a. We’ll prove here the converse of this observation, in other words we’ll prove