Title: Polynomials, Ellipses, and Matrices: Two Questions, One Answer
Abstract: AbstractConsider the following questions for points a1, a2 in the unit disc, 픻. If q(z) = (z − a1)(z − a2), when is q the derivative of a polynomial with all of its zeros on the unit circle, ∂픻? If an ellipse E with foci a1, a2 is inscribed in a triangle with vertices on ∂픻, when is E tangent at the midpoints to a triangle with vertices on ∂픻? We show that these problems are essentially the same. In fact, the answer to both is a very simple: if and only if 2❘a1a2❘ = ❘a1 + a2❘. We also discuss generalizations of these problems and their solutions. Additional informationNotes on contributorsPamela GorkinPAMELA GORKIN received her B.S., M.S., and Ph.D. from Michigan State University. She also spent a year at Indiana University where she learned about numerical ranges from Paul Halmos. She has been teaching at Bucknell University since 1982, with time off for good behavior. Her hobbies are hiking, reading, traveling, cooking and eating, though not necessarily in that order.Elizabeth SkubakELIZABETH SKUBAK attended Bucknell University for her undergraduate studies and is currently a graduate student and teaching assistant at the University of Wisconsin-Madison. She likes to spend her few waking, non-working hours reading, cooking, taking photographs, and being outside.
Publication Year: 2011
Publication Date: 2011-01-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 13
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