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0. Introduction 1. Holomorphic nondegenera~y of real-analytic manifolds 1.1. Preliminaries on real submanifolds of C N 1.2. Holomorphic nondegeneracy and its propagation 1.3. The Levi number and essential finiteness 1.4. Holomorphic nondegeneracy of real algebraic sets 2. The Segre sets of a real~analytic CR submanifold 2.1. Complexification of M, involution, and projections 2.2. Definition of the Segre sets of M at p0 2.3. Homogeneous submanifolds of CR dimension 1 2.4. Homogeneous submanifolds of arbitrary CR dimension 2.5. Proof of Theorem 2.2.1 3. Algebraic properties of holomorphic mappings between real algebraic sets 3.1. A generalization of Theorems 1 and 4 3.2. Propagation of algebralcity 3.3. Proof of Theorem 3.1.2 3.4. Proof of Theorem 3.1.8 3.5. An example 3.6. Proofs of Theorems 1 through 4

Algebraicity of holomorphic mappings between real algebraic sets in Cn