A Zero Structure Theorem for Differential Parametric Systems

✍ Scribed by Xiao-Shan Gao; Shang-Ching Chou

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 273 KB
• Volume: 16
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.1993.1065

✦ Synopsis

We present a zero structure theorem for a differential parametric system:

[
p_{1}=0, \cdots, p_{r}=0, d_{1} \neq 0, \cdots, d_{s} \neq 0
]

where (p_{i}) and (d_{i}) are differential polynomials in (K\left{u_{1}, \cdots, u_{m}, x_{1}, \cdots, x_{n}\right}) and the (u) are parameters. According to this theorem we can identify all parametric values for which the parametric system has solutions for the (x_{i}) and at the same time giving the solutions for the (x_{i}) in an explicit way, i.e., the solutions are given by differential polynomial sets in triangular form. In the algebraic case, i.e. when (p_{i}) and (d_{i}) 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.