Effective Differential Nullstellensatz for Ordinary DAE Systems with Constant Coefficients

TL;DR

This paper establishes explicit upper bounds for the differential Nullstellensatz in ordinary differential algebraic systems with constant coefficients, improving understanding of solution ideal membership.

Contribution

It provides the first explicit, non-elementary recursive bounds for the differential Nullstellensatz in systems with constant coefficients.

Findings

Derived bounds for ideal membership involving derivatives up to order L

Bound for M indicating algebraic ideal membership of f^M

Bounds are non-elementary recursive, improving previous results

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.