Geometric Fiber Classification of Morphisms and a Geometric Approach to Cylindrical Algebraic Decomposition

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.