Effective Differential Nullstellensatz for Ordinary DAE Systems with Constant Coefficients

Original · EN

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 x be a set of n 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 K. Let d:={°(f), °(f)} and ε:={2,ord(f), ord(f)}. We show that fᵐ belongs to the algebraic ideal generated by the successive derivatives of f of order at most L = (nεd)²ᶜ⁽ⁿε⁾³, for a suitable universal constant c>0, and M=dⁿ⁽ε⁺ˡ⁺¹⁾. The previously known bounds for L and M are not elementary recursive.