Project Description: |
Real and non-real zeros
There are many open questions dealing with the zeros and zeros of derivatives of real polynomials and functions of the form P(z)eQ(z) where P and Q are real polynomials. One such questions deals with the number of points of extreme curvature of a polynomial. Another asks about the number and location of non-real zeros of the derivatives of P(z)eQ(z). And finally, one asks about the relationship between the number of non-real zeros of a polynomial and the number of critical points of the logarithmic derivative of the polynomial. The second and third problems listed have roots in Pólya and Gauss respectively, so they are very old. This summer we will be looking at those three questions using a geometrical level curves technique. In particular, we will look at the level sets {z
in H+:Im f(z)=0} and {z in H+:Re f(z)=0}.
Background Needed: sophomore level linear algebra, multivariable calculus, some experience with complex numbers, some familiarity with programming in Maple or Mathematica. Knowledge of real and complex analysis would be helpful, but not essential.
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