Mentor: Professor Robert Wilson
What I've Studied So Far
During the first two weeks of my project, I studied the method of partial fractions in the noncommutative case. I wrote a proof that shows it is indeed possible to generalize the method of partial fractions to the noncommutative case. The conjecture discussed in my PowerPoint presentation on June 11th about the coefficients in the generalized method of partial fractions is true.
Here is my presentation:
I next generalized to the noncommutative case Theorem 1.18 from Holtz and Tyaglov's "Structured Matrices, Continued Fractions, and Root Localization of Polynomials". The reference for this paper is listed below. My generalization shows a connection between Laurent series and quasideterminants. References regarding the theory of quasideterminants are listed below.
I also studied Galois's result about periodic continued fractions and quadratic equations. Galois showed that every periodic continued fraction over a field satisfies a quadratic equation. I generalized this result for periodic continued fractions over division rings. I found that every periodic continued fraction over a division ring satisfies xAx + Bx + xC +D = 0, where A, B, C, and D are elements of the division ring. The A, B, C, and D can be written explicitly in terms of the repeating terms of the corresponding periodic continued fraction.
Galois showed that there is a relationship between a periodic continued fraction and the periodic continued fraction obtained when the order of the repeating terms is reversed. I attempted to generalize this result, but was unable to write the two continued fractions in terms of each other.
Here is my second presentation:
What I'm Currently Studying
I am returning to my proof about partial fraction decomposition. I would like to see if the same results can be found without use of the Gelfand-Retakh Vieta Theorem.
Gel'fand, I. M.; Retakh, V. S. Theory of noncommutative determinants, and characteristic functions of graphs. (Russian) Funktsional. Anal. i Prilozhen. 26 (1992), no. 4, 1--20, 96; translation in Funct. Anal. Appl. 26 (1992), no. 4, 231--246 (1993)
Holtz, Olga, and Mikhail Tyaglov. "Structured Matrices, Continued Fractions, and Root Localization of Polynomials.” http://www.cs.berkeley.edu/~oholtz/RF.pdf
Lauritzen, Neils. "Continued Fractions and Factoring.”
Wilson, Robert L. "Three Lectures on Quasideterminants." Lecture. http://www.mat.ufg.br/bienal/2006/mini/wilson.pdf