Noncommutative Partial Fractions and Continued Fractions

1. Noncommutative Partial Fractions and Continued Fractions Darlayne Addabbo Professor Robert Wilson Department of Mathematics Rutgers University July 16, 2010

2. Overview • Existence of partial fraction decompositions over division rings • Relation between Laurent series and quasideterminants • Generalization of Galois’s result that every periodic continued fraction satisfies a quadratic equation

3. Partial Fractions in C (complex numbers) • If f(x) is a polynomial over C of degree n with distinct roots , then for some in C.

4. Example

5. We may also write as

6. The two expressions are the same over C. However, over a division ring, is ambiguous. (It could be equal to or .) To avoid ambiguity, we prefer the second notation.

7. Can we generalize the method of partial fractions to an arbitrary division ring, D? • Recall that a division ring satisfies all of the axioms of a field except that multiplication is not required to be commutative. • Over a field, if f(x) is a monic polynomial of degree n, with n distinct roots , • But this doesn’t work in a division ring. It doesn’t even work in the quaternions.

8. (Recall) The Algebra of Quaternions • The algebra of quaternions is a four dimensional vector space over R (the real numbers), with basis 1, i, j, k and multiplication satisfying: • ij=-ji=k • jk=-kj=i • ki=-ik=j • 1 is the multiplicative identity

9. An example has roots i+1 and 1+i+j Check: Similarly,

10. But Instead, Note that each root corresponds to the rightmost factor of one of these expressions. (This is due to a result of Ore.)

11. Using the above to solve partial fractions where , are elements of the quaternions

12. So which gives and (since we can write and in terms of 1+i and 1+i+j), Thus So we have generalized the method of partial fractions.

13. Theorem Let , with , have distinct roots, . Then there exist elements, such that

14. First prove inductively that

15. 2) By the Gelfand-Retakh Vieta Theorem, we can write where each is an element of our division ring, D, and for all

16. 3) Obtaining our set of equations Recall that we want to find such that We multiply both sides of our equation by f(x). Notice that our terms cancel.

17. 4) Obtaining our System of Equations Comparing terms on both sides of our equation, we obtain Using the Generalized Cramer’s Rule, we can solve this system of equations .

18. Continued Fractions In the commutative case, periodic continued fractions are often written as follows , a field.

19. Continued Fractions In the noncommutative case, we avoid ambiguity by writing our continued fractions using nested parentheses. where , a division ring

20. There is a theorem by Galois which says that every periodic continued fraction is a solution to a quadratic equation. I generalized this to show that a periodic continued fraction over a division ring, D, satisfies where We can write in terms of , the repeating terms of our continued fraction

21. Current Work Galois showed that there is a relationship between the complex conjugate of a periodic continued fraction and the periodic continued fraction obtained when we write the repeating terms in reverse order. We want to generalize this to the non-commutative case.

22. References Gelʹfand, 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.” http://www.dm.unito.it/~cerruti/ac/cfracfact.pdf Wilson, Robert L. "Three Lectures on Quasideterminants." Lecture. http://www.mat.ufg.br/bienal/2006/mini/wilson.pdf