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SECTOR ANALOGUE OF THE GAUSS-LUCAS THEOREM

2019
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Canadian Journal of Mathematics - Journal Canadien de Mathematiques
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The classical Gauss-Lucas theorem states that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. This fundamental theorem follows from the fact that if all the zeros of a polynomial are in a half plane, then the same is true for its critical points. The main result of this work replaces the half plane with a sector as follows. We show that if the coefficients of a monic polynomial p(z) are in the sector {te iψ : ψ ∈ [0, φ], t ≥ 0}, for some φ ∈

doi:10.4153/s0008414x19000609
fatcat:d3cgr2gjuzcrddqkq42hsiw4wm