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We give upper bounds for the differential Nullstellensatz in the case ofordinary systems of differential algebraic equations over any field ofconstants $K$ of characteristic $0$. Let $\vec{x}$ be a set of $n$ differentialvariables, $\vec{f}$ a finite family of differential polynomials in the ring$K\{\vec{x}\}$ and $f\in K\{\vec{x}\}$ another polynomial which vanishes atevery solution of the differential equation system $\vec{f}=0$ in anydifferentially closed field containing $K$. Let $d:=\max\{\deg(\vec{f}),\deg(f)\}$ and $\epsilon:=\max\{2,{\rm{ord}}(\vec{f}), {\rm{ord}}(f)\}$. Weshow that $f^M$ belongs to the algebraic ideal generated by the successivederivatives of $\vec{f}$ of order at most $L = (n\epsilond)^{2^{c(n\epsilon)^3}}$, for a suitable universal constant $c>0$, and$M=d^{n(\epsilon +L+1)}$. The previously known bounds for $L$ and $M$ are notelementary recursive.

Effective Differential Nullstellensatz for Ordinary DAE Systems with Constant Coefficients