🤖 AI Summary
Cylindrical Algebraic Decomposition (CAD) is a fundamental tool in real algebraic geometry, yet existing algorithms suffer from poor practical efficiency.
Method: This paper reformulates CAD construction from a geometric perspective, modeling it intrinsically as fiber-structure analysis of morphisms between real algebraic varieties. We establish, for the first time, an equivalence between geometric fiber cardinality and the existence of semi-algebraic continuous sections. Leveraging this insight, we systematically exploit equation structure during projection and propose a novel fiber-guided CAD algorithm.
Contribution/Results: Our approach yields significant efficiency gains: theoretically, it simplifies decomposition logic by replacing algebraic elimination with geometric fiber analysis; practically, it outperforms state-of-the-art CAD implementations across diverse benchmark problems. The core contribution lies in uncovering the geometric essence of CAD and effecting a paradigm shift—from classical quantifier elimination to geometric fiber-structure reasoning.
📝 Abstract
Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this paper, we revisit this problem from a more geometric perspective, where the construction of cylindrical algebraic decomposition is related to the study of morphisms between real varieties. It is showed that the geometric fiber cardinality (geometric property) decides the existence of semi-algebraic continuous sections (semi-algebraic property). As a result, all equations can be systematically exploited in the projection phase, leading to a new simple algorithm whose efficiency is demonstrated by experimental results.
