Effective differential Nullstellensatz for ordinary DAE systems with constant coefficients

We give upper bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over the field of complex numbers. Let x be a set of n differential variables, f a finite family of differential polynomials in the ring C{x} and f ∈ C{x} another polynomial which vanishes at every solution of the differential equation system f = 0 in any differentially closed field containing C. Let d := max{deg(f), deg(f)} and ǫ := max{2, ord(f), ord(f)}. Then, f belongs to the algebraic ideal generated by the successive derivatives of f of order at most L = (nǫd) c(nǫ)3 , for a suitable universal constant c > 0, and M = d. The previously known bound for the number L of required differentiations is described in terms of the Ackermann function and, in particular, it is not primitive recursive.