Generalizing Continued Fractions Darlayne Addabbo Professor Robert Wilson Department of Mathematics Rutgers University June 11, 2010

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

Example Here and

Can we generalize this process to arbitrary division rings? 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.

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

An example of the difficulties found in working over the quaternions has roots i+1 and 1+i+j Check:

But However, Note that each of the above factorizations contains a multiplicative of where is a root of f(x). This is due to the Gelfand-Retakh Vieta Theorem

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

So which gives and (since we can write and in terms of 1+i and 1+i+j), We can use the Generalized Cramer’s Rule to solve this set of equations. In this case So we have generalized the method of partial fractions.

Our Conjecture Given a polynomial of degree n, we can generalize the above and obtain a system of equations: which can be solved using the generalized Cramer’s Rule