On The Zeros of Some Complex harmonic polynomials

It is well known that the Fundamental Theorem of Algebra doesn’t hold true in the case of complex harmonic polynomials. The motive of the manuscript is to explore the zeros of a complex harmonic polynomial F(z) of degree \((n+m)\) $(n+m)$ . In particular, we prove that the sum of the orders of the zeros of F(z) is \(-(n+m)\) $-(n+m)$ . We also show that F(z) has a total \((n+m+2)\) $(n+m+2)$ or \((n+m+2k+2)\) $(n+m+2k+2)$ number of zeros under different conditions on the coefficients. In addition to the count of zeros, we locate the zeros of F(z) and obtain that all non-trivial zeros of F(z) lie in the given annular region.