Type: Article
Publication Date: 1998-01-01
Citations: 28
DOI: https://doi.org/10.1090/s0002-9939-98-04381-0
The classical Gauss-Lucas Theorem states that all the critical points (zeros of the derivative) of a nonconstant polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lie in the convex hull <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Xi"> <mml:semantics> <mml:mi mathvariant="normal">Ξ</mml:mi> <mml:annotation encoding="application/x-tex">\Xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the zeros of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is proved that, actually, a subdomain of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Xi"> <mml:semantics> <mml:mi mathvariant="normal">Ξ</mml:mi> <mml:annotation encoding="application/x-tex">\Xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains the critical points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.