The critical values of a polynomial
It has been known that any real sequence \(y_1,y_2,\dots, y_n\) is the sequence of the critical values of some real polynomial of degree \(n+1\). In this paper it is shown that any complex sequence \(w_1,w_2,\dots, w_n\) is the sequence of the critical values of some complex polynomial of degree \(n+1\). For any vector \(u= (u_1,\dots, u_n)\in \mathbb{C}^n\), the polynomial \[ p_u(z)= \int^z_0 (w- u_1)\cdots(w- u_n) dw \] has the critical values \[ v= (v_1,\dots, v_n)= \Theta(u)= \Theta(u_1,\dots, u_n)= (p_u(u_1),\dots, p_u(u_n)). \] The map \(\Theta(u)\), \(\mathbb{C}^n\to \mathbb{C}^n\), defined above is holomorphic, open, closed, proper and finite. Further, for each \(v\in \mathbb{C}^n\) there are exactly \((n+1)^n\) points \(u\) such that \(\Theta(u)= v\).
scientific article; zbMATH DE number 1885725