We study the critical points of a complex cubic polynomial, normalized to have the form p(z)= (z - 1)(z - r1)(z - r2) with |r1| = 1 = |r2|. If T_γ denotes the circle of diameter γ passing through 1 and 1 - γ, then there are α,β ∈[ 0,2] such that one critical point of p lies on T_α and the other on T_β. We show that T_β is point of such a polynomial almost always determines the polynomial uniquely, and (2) there is a “desert” in the unit disk, the open disk {z ∈ C:|z-2/3|﹤1/3}in which critical points cannot occur.
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