Pressing Toward the Prize

Critical Points of Polynomials

Posted on: October 13, 2010

On October 6, we had a guest speaker in Capstone, Dr. Julie Eaton from the University of Puget Sound, who gave a presentation on locating critical points of polynomials. First, she presented the Lucas-Gauss Theorem: “The roots of the derivative of polynomials are contained in the convex hull of the roots of that polynomial.” For polynomials with real roots, it was easy to see graphically that the roots of the derivative, which are the critical points of the continuous polynomial curve, would be contained within the same interval as the roots of the polynomial. I had a little more difficulty following what was happening when some of the roots were complex numbers, however, because none of my classes to date have covered graphing in the complex number system. I know that when the root of a polynomial is a complex number, its conjugate is also a root. This made it easier to understand that the graph containing complex roots could be connected to form a region rather than an interval. It was interesting to see that the same principle in the real number system was true in the complex numbers. The region created by the roots of the derivatives was fully contained in the roots of the polynomial. For higher order polynomials, each time a derivative was taken, the shape of the region “decreased” and was still fully contained. For example, a fifth degree polynomial with one real and four complex roots created a pentagon, the first derivative created a quadrilateral, the second a triangle, and so on, each fully contained within the other. One of the things I love about math is finding patterns in “unexpected” places, so this was fascinating to me.

The second concept Dr. Eaton covered had to do with Newton polygons, which Newton created in 1671 along with an algorithm used to approximate the roots of polynomials as functions of their coefficients. I had no difficulty in actually using the technique to approximate the roots, but I didn’t fully understand the explanation of why it works. I guess that will wait for another day! I found the presentation quite interesting, but it went a little “fast” for me. I would like to have had more time to explore with Dr. Eaton a few of the concepts that were new to me, but she was under specific time constraints, and it was a lecture, after all. For me, the best part of the presentation is that, even though I didn’t fully understand everything she shared, it gave me ideas on new avenues of study.

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1 Response to "Critical Points of Polynomials"

Very nice review of the talk. I think, based on what you said above, that the talk was a success for you.

You got the main point I think: there are interesting, accessible, math problems out there that you could study or think about if you found yourself with extra time and interest. It is always nice to know what has been done and what people find interesting enough to talk about.

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