rectification; Abhyankar-Sathaye embedding problem; variable real algebraic geometry; real algebraic varieties; complexification; Smith's theory; Galois-Maximal varieties; algebraic cycles; real algebraic models; algebraic curves; algebraic surfaces; topology of algebraic varieties; regular maps; rational maps; singularities; algebraic approximation; Comessatti theorem; Rokhlin theorem; Nash conjecture; Hilbert's XVI problem; Cremona group; real fake planes 