Roots of a Polynomial and its Derivative -- Patrick Gallagher
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Four ways the roots of f ' are related to the roots of f:
<br> 1) Gauss, Lucas (1836,1879): The roots of f' are contained in the smallest convex set containing the roots of f;
<br> 2) Laguerre? (1880's?): For f real on R, f' has no more nonreal roots than f;
<br> 3) Jensen (1912): For f real on R, each nonreal root of f' is contained in at least one "Jensen disc", i.e. a closed disc with with diameter between a pair of conjugate nonreal roots of f.
<br> 4) M. Riesz (1927): For f real on R, if f has only real & simple roots, then the same for f'; if also deg f > 2, then the minimal distance between distinct roots of f' exceeds the minimal distance between distinct roots of f.