This thesis investigates the zeros of entire harmonic mappings on Rn. The starting point of this work is the failure to generalize Bézout’s theorem to entire holomorphic mappings, known as the Transcendental Bézout Problem, due to the instability of classical zero counting highlighted by the Cornalba and Shiffman counterexample [CS72]. Drawing inspiration from recent work in [Buh et al., 24b] on the Persistent Transcendental Bézout Theorem, and from the real harmonic analogue of the Cornalba and Shiffman counterexample presented in [Sto24], we extend their approach to this setting, a research direction explicitly suggested in [Buh et al., 24b, section 1.5.3]. The main objective is to obtain a bound on the growth rate of the number of connected components containing at least one zero within a ball as a function of its radius. We prove that this number is polynomially bounded by the growth rate of the maximum modulus of the mapping. The proof adapts the coarse nodal counting techniques of [Buh et al., 24a] and the methodology of [Buh et al., 24b]. The first chapter presents the properties of harmonic mappings and establishes Cauchytype estimates. The second chapter introduces a simplified version of stratified Morse theory adapted to manifolds with corners. This allows relating the local behavior of the mapping to the global topology of its sublevel sets on the closed ball via critical points. The third chapter presents persistence and its stability theorem. The final chapter proves the main result by combining these tools. Polynomial approximation allows applying Milnor’s algebraic complexity bounds, while persistent homology ensures the transfer of these bounds to the transcendental mapping.