Here at PSU, Sergei Tabachnikov just finished giving a two-week mini-course on the “geometry of polynomials.” The collection of topics was diverse: various proofs of the fundamental theorem of algebra, resultant and discriminants, Chebyshev polynomials, harmonic functions in three-space, and a sketch of the proof of the four-vertex theorem. The lectures presupposed familiarity with no more than elementary analysis and linear algebra, though more advanced topics were referenced (without proofs).

Two weeks really means about seven days; class was cancelled because apparently people thought that most of us were interested in an art show.

For the benefit of the huddled masses yearning to be educated, here are the notes that I took from these lectures. The file is rather large (40 MB) because of the insertion of jpg images that someone else in the REU drew. The djvu file is a lot smaller, but WordPress won’t let me post it, so email me if you want it.

The notes are mostly a faithful representation of what I took in class, but I have edited them lightly to moderate my tendency to embarrass myself.

Next week, Yakov Pesin is lecturing on fractal geometry and dynamics; I’ll post those notes when I’m done with them.

### Like this:

Like Loading...

*Related*

July 17, 2010 at 7:36 am

Till now I have only read a small part of your notes. Thanks for having made your notes available. I see you are doing live-ing efficiently, since you “felt behind” only once ): .

A. Using the notation of the 2001 Victor Prasilov’s book

Polynomials(Springer), Chebyshev polynomials are defined asfor

and the monic polynomials as

The following result is proved:

, where is an odd prime.In one of the exercises readers are asked to prove:

The function satisfies the differential equation.

B. Could you please tell me which environment do you use to insert figures and how?

July 17, 2010 at 11:30 am

A. Hm, interesting, thanks for mentioning. The first one should be an application of the recurrence relation of the Chebyshev polynomials, but I have to play around with it a bit more to get it into the right form. The second follows from the explicit formula for the Chebyshev polynomials (Thm 7.5) and the fact that the binomial coefficients are divisible by (and also Fermat’s little theorem).

B. I use \begin{figure} \includegraphics*[width=10cm]{filename.jpg} \end{figure} (which requires the graphicx package). I actually don’t know too many other ways to do it…

July 17, 2010 at 2:15 pm

In my first comment, the smiley should have been a smiley face. Please excuse me for this silly mistake.

Thanks for your reply to A.

Concerning B, I use Scientific Work Place 4.10 to create LaTeX documents easily and generate graphics, but to solve my problem most likely I have to install a “standard” TeX system and the package you indicate.

July 17, 2010 at 2:30 pm

Hm, I see. I’m just using LaTeX pretty much as it is (i.e., with emacs). I perused the Scientific Workplace website (having never used their products), but it does seem to claim to be able to handle figures. But, at any rate, if there’s a problem, one can of course simply use SW to edit the files, place \begin{figure}<more code. \end{figure}, and then run pdflatex in an ordinary shell with graphicx.sty in the appropriate directory :-).