On the number of zeros of  harmonic polynomials arranged in alternating  patterns

In this paper, we investigate the number of zeros of harmonic polynomials of the form$$f(z)=a_{1}z^{n_{1}}+a_{2}\overline{z}^{n_{2}}+a_{3}z^{n_{3}}+\cdots+\left(\frac{1+(-1)^{k+1}}{2}a_{k}z^{n_{k}}+\frac{1+(-1)^{k}}{2}a_{k}\overline{z}^{n_{k}}\right)+a_{k+1},$$where $a_{j}~(j=1,2\cdots,k+1)$ are non-zero constants and $n_{j}~(j=1,2\cdots,k)$ are positive integers with $n_{1}>n_{2}>\cdots>n_{k}\geq1$.Assume that $f$ is regular. Then we show that $f$ has at least $n_1+2(n_2+n_3+\cdots+n_k)$ zeros when each of its corresponding real polynomials has two distinct positive zeros.Furthermore, there are some such harmonic polynomials which have at least $n_1+n_2(n_2+1)$ zeros under particular conditions.In particular, if we choose $n_1=n$ and $n_2=n-1$, then it provides an array of new examples for the sharp bound $n^2$ of zeros of harmonic polynomials $p+\overline{q}$,where $p$ and $q$ are analytic polynomials with degree $n$ and $n-1$ respectively.The sharpness in case $n=3$ is continued to studied, and thus a sufficient condition on two parameters $a$ and $b$ is obtained by Cardano's Formulafor which the harmonic polynomial $f(z)=z^3+a\overline{z}^2+bz+1$ has exactly 9 zeros.