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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 1,..., n , then 1 a1 an ... f (x) x 1 x n for some a1,...,an in C. Example 1 a1 a2 2 x 1 x x1 x x 2 1 Here a1 a2 and x1 x2 1 2 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 1,..., n , f (x) (x 1)...(x n ) • 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 f (x) x (2 j)x (2 j k) 2 has roots i+1 and 1+i+j 2 (i 1) (2 j)(i 1) 2 j k Check: 1 2i 1 2i k 2 j 2 j k 0 But f (x) (x (1 i j))(x (1 i)) However, f (x) (x (1 i j))(x (1 i)) (x (1 i))(x (1 i j)) Note that each of the above factorizations contains a multiplicative of x i where i is a root of f(x). This is due to the Gelfand-Retakh Vieta Theorem Using the above to solve partial fractions (x 2 (2 j)x (2 j k))1 (x (1 i))1 a1 (x (1 i j))1 a2 x 2 (2 j)x (2 j k) (x y 2 )(x (1 i)) (x z2 )(x (1 i j)) where y 2,z2 are elements of the quaternions So 1 (x y2 )a1 (x z2 )a2 which gives a1 a2 0 and (since we can write y2 and z 2 in terms of 1+i and 1+i+j), (1 i)a1 (1 i j)a2 1 Generalized We can use the Cramer’s Rule to solve this set of equations. In this case a1 j a2 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: 1 1 n1 1 1 2 n1 2 1 a1 0 n a2 0 ... n1 an 1 n ... ... which can be solved using the generalized Cramer’s Rule