Summary: In this paper, we study the equation \(z^n=\sum_{k=0}^{n-1} a_k z^k\), where \(\sum_{k=0}^{n-1}a_k =1\), \(a_k\geq 0\) for each \(k\). We show that, given \(p>1\), there exist \(C(1/p)\)-polynomials with the degree of weighted sum \(n-1\). However, we obtain sufficient conditions for nonexistence of certain lacunary \(C(1/p)\)-polynomials. In case of the degree of weighted sum \(n-2\), we see that, by giving an example, our sufficient condition is best possible in a certain sense.