Effective Differential Nullstellensatz for Ordinary DAE Systems with Constant Coefficients

Abstract: 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 $\vec{x}$ be a set of $n$ differential variables, $\vec{f}$ a finite family of differential polynomials in the ring $K{\vec{x}}$ and $f\in K{\vec{x}}$ another polynomial which vanishes at every solution of the differential equation system $\vec{f}=0$ in any differentially closed field containing $K$. Let $d:=\max{\deg(\vec{f}), \deg(f)}$ and $\epsilon:=\max{2,{\rm{ord}}(\vec{f}), {\rm{ord}}(f)}$. We show that $f^M$ belongs to the algebraic ideal generated by the successive derivatives of $\vec{f}$ of order at most $L = (n\epsilon d)^{2^{c(n\epsilon)^3}}$, for a suitable universal constant $c>0$, and $M=d^{n(\epsilon +L+1)}$. The previously known bounds for $L$ and $M$ are not elementary recursive.