A zero-dimensional approach to compute real radicals

• Main Author: Silke J. Spang
• Format: Article
• Language: English
• Published: Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova 2008-04-01
• Series: Computer Science Journal of Moldova
• Online Access: http://www.math.md/files/csjm/v16-n1/v16-n1-(pp64-92).pdf

The notion of real radicals is a fundamental tool in Real Algebraic Geometry. It takes the role of the radical ideal in Complex Algebraic Geometry. In this article I shall describe the zero-dimensional approach and efficiency improvement I have found during the work on my diploma thesis at the University of Kaiserslautern (cf. [6]). The main focus of this article is on maximal ideals and the properties they have to fulfil to be real. New theorems and properties about maximal ideals are introduced which yield an heuristic prepare_max which splits the maximal ideals into three classes, namely real, not real and the class where we can't be sure whether they are real or not. For the latter we have to apply a coordinate change into general position until we are sure about realness. Finally this constructs a randomized algorithm for real radicals. The underlying theorems and algorithms are described in detail.