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 KK of characteristic 0. Let x be a set of nn differential variables, f a finite family of differential polynomials in the ring K{x} and f∈K{x} another polynomial which vanishes at every solution of the differential equation system f=0 in any differentially closed field containing KK. Let d≔max{deg(f),deg(f)} and ϵ≔max{2,ord(f),ord(f)}. We show that fMfM belongs to the algebraic ideal generated by the successive derivatives of f of order at most L=(nϵd)2c(nϵ)3L=(nϵd)2c(nϵ)3, for a suitable universal constant c>0c>0, and M=dn(ϵ+L+1)M=dn(ϵ+L+1). The previously known bounds for LL and MM are not elementary recursive.