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 any field of constants K of characteristic 0. Let x be a set of n differential variables, f a finite family of differential polynomials in the ring K{x} and f is an element of K{x} another polynomial which vanishes at every solution of the differential equation system f = 0 in any differentially closed field containing K. Let d := max{deg(f), deg(f)} and is an element of := max{2, ord(f), ord(f)). We show that f(M) belongs to the algebraic ideal generated by the successive derivatives off of order at most L = (n is an element of d)2(c(n is an element of)3), for a suitable universal constant c > 0, and M = d(n(is an element of+L+1)) The previously known bounds for L and M are not elementary recursive. (C) 2014 Elsevier Inc. All rights reserved.