The Tricomplex Polynomial and Its Root Structure

This paper develops a systematic framework for the study of tricomplex polynomials and their roots within the multicomplex space C_3. A complete characterization of root behavior is established, encompassing all possible cases—finitely many roots, no roots, and infinitely many roots. Under the non-singularity condition γ_(n )∉O(i_1,i_2,i_3), we show that every tricomplex polynomial factors fully into linear components. Consequently, we prove that a polynomial of degree n in the tricomplex space C_3 possesses exactly n^4 roots, counted with multiplicities, thereby extending the classical Fundamental Theorem of Algebra to the tricomplex setting. We further identify conditions ensuring that the presence of one root guarantees all of its conjugates as roots. Additionally, the quadratic equation ζ^2=η, for η∈C_3, is solved in complete generality, yielding a full classification of tricomplex square roots. These results enhance the structural understanding of polynomial theory in tricomplex algebra and contribute to the broader development of multicomplex analysis.