Research Projects

Since 1984, there has been extensive interest in univalent harmonic mappings, with complex-valued harmonic functions being a natural generalization to analytic functions. Research has explored similarities and differences between these functions, with important results like Argument Principle, Rouche's Theorem, and Hurwitz's theorem still valid for complex harmonic functions. However, many basic questions and conjectures remain unresolved, such as the zeros of complex-valued harmonic polynomials. The Fundamental Theorem of Algebra (FTA) states that every analytic polynomial of degree n has exactly n zeros in the complex plane, but does not extend to complex-valued harmonic polynomials. The problem of finding the maximal count of zeros of complex-valued harmonic polynomials is an interesting open problem, as it appears in various areas with different levels of difficulty. The primary objective of this proposal is to find the exact count of zeros and placement of zeros of some harmonic polynomials and trinomials.