In this paper, the authors construct a polynomial which has an important property in the theory of complex dynamics. Let \(p\) be a polynomial. Define the associate relaxed Newton map as \[ N_{p,h}: \hat{\mathbb C} \to \hat{\mathbb C},\;\;N_{p,h}(z)=z-h\frac{p(z)}{p'(z)}, \] where \(h\in\mathbb{D}_1(1)\) is a complex parameter. Here \(\mathbb{D}_1(1)\) denotes the open disk of radius \(1\) centered at \(1\). The main result is the following:
Let \(\Omega=\{z_1, z_2, \dots , z_n\}\) be any \(n\) distinct points in the complex plane, \(n\geq 2\). Then for any \(h\in\mathbb{D}_1(1)\) there exists a polynomial \(p\) of degree at most \(n+1\) so that \(\Omega\) is a super-attracting cycle for \(N_{p,h}\). It is also showed that their estimates are sharp in degree, namely, such Newton map with a super-attracting cycle must have degree at least \(n+1\). This result is an improvement for the results of S. Plaza and V. Vergara [Sci., Ser. A, Math. Sci. (N.S.) 7, 31–36 (2002; Zbl 1137.37317)] and S. Plaza and N. Romero [J. Comput. Appl. Math. 235, No. 10, 3238–3244 (2011; Zbl 1215.65089)]. Some examples and graphics are given as well.