Cylindrical Algebraic Decomposition (CAD) is a classical construction in real algebraic geometry. The original cylindrical algebraic decomposition was proposed by Collins, using the classical elimination theory. In this paper, we first study the geometric fibers cardinality classification problem of morphisms of affine varieties (over a field of characteristic 0), using a constructive version of Grothendieck's Generic Freeness Lemma and Parametric Hermite Quadratic Forms, then we show how cylindrical algebraic decomposition is related to this classification problem. This provides a new geometric view of Cylindrical Algebraic Decomposition and a new theory of Cylindrical Algebraic Decomposition is developed in this paper.

翻译：柱形代数分解（CAD）是实代数几何中的经典构造。最初的柱形代数分解由Collins提出，采用了经典消元理论。本文首先利用Grothendieck一般自由性引理的构造性版本及参数化Hermite二次型，研究仿射簇态射的几何纤维基数分类问题（特征为零域上），进而揭示柱形代数分解与该分类问题的关联。由此，本文建立了柱形代数分解的新几何视角，并发展了柱形代数分解的新理论。