A Zero Structure Theorem for Differential Parametric Systems

We present a zero structure theorem for a differential parametric system: p1 = 0, · · · , pr = 0, d1 6= 0, · · · , ds 6= 0 where pi and di are differential polynomials in K{u1, · · · , um, x1, · · · , xn} and the u are parameters. According to this theorem we can identify all parametric values for which the parametric system has solutions for the xi and at the same time giving the solutions for the xi in an explicit way, i.e., the solutions are given by differential polynomial sets in triangular form. In the algebraic case, i.e. when pi and di are polynomials, we present a refined algorithm with higher efficiency. As an application of the zero structure theorem presented in this paper, we give a new algorithm of quantifier elimination over differential algebraic closed fields. The algorithm has been implemented and several examples reported in this paper show that the algorithm is of practical value.