Effective differential Nullstellensatz for ordinary DAE systems with constant coefficients - D'Alfonso, Lisi (2014)
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 Kx and fεKx another polynomial which vanishes at every solution of the differential equation system f=0 in any differentially closed field containing K. Let d max{deg(f),deg(f)} and max{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=(nd)2c(n)3, for a suitable universal constant c>0, and M=dn(+L+1). The previously known bounds for L and M are not elementary recursive.