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.

翻译：圆柱代数分解是实代数几何中的经典构造。尽管存在多种计算圆柱代数分解的算法，其实际性能仍十分有限。本文从更几何化的视角重新审视该问题，将圆柱代数分解的构造与实代数簇间态射的研究相联系。研究表明几何纤维基数（几何性质）决定了半代数连续截面的存在性（半代数性质）。由此，所有方程可在投影阶段被系统化利用，从而提出一种新型简洁算法，其有效性已通过实验结果得到验证。