Year: 2013
Type: preprint
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 $d:=\max\{°(\vec{f}), °(f)\}$ and $ε:=\max\{2,{\rm{ord}}(\vec{f}), {\rm{ord}}(f)\}$. We show that $f^M$ belongs to the algebraic ideal generated by the successive derivatives of $\vec{f}$ of order at most $L = (nεd)^{2^{c(nε)^3}}$, for a suitable universal constant $c>0$, and $M=d^{n(ε+L+1)}$. The previously known bounds for $L$ and $M$ are not elementary recursive.
Source: arXiv (Cornell University)
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G2867063790, G3386563612, G4273107380
OpenAlex ID: W1984036896