Asian Research Journal of Mathematics

This paper presents a systematic study of the zeros of bicomplex meromorphic functions on the bicomplex plane T. Via the idempotent decomposition, every bicomplex meromorphic function f corresponds to a pair of complex meromorphic functions (f₁, f₂). We introduce the notions of strong zeros, at which both component functions vanish to positive order, and weak zeros, at which exactly one component vanishes. The set of strong zeros of f is characterized precisely as the collection of points w = αe₁ + βe₂ for which both f₁(α) = 0 and f₂(β) = 0. For strong zeros we establish a local factorization theorem in which the multiplicity, termed the bidegree, is defined as the minimum of the vanishing orders of the two components. Building on this foundation, we derive bicomplex analogues of several classical results: an Enestr¨om–Kakeya-type theorem that confines the zeros to a componentwise exterior region E(1; 1, 1), a Gauss–Lucas-type theorem asserting that the zeros of the derivative lie in the idempotent convex hull of the zeros of f, and a Rouch´e-type theorem for counting zeros in product domains. We further show that the zero-counting function in a bicomplex disk factors as the product of the counting functions of the two complex components, and that the exponent of convergence of the zeros of a bicomplex entire function of finite order equals the maximum of the corresponding component exponents.