Let phi be rational given on the unit circle \z\ = 1 by phi = c(0)Pi (p)(j=1)(z-nu (j))/Pi (h)(j=1)(1-z/rho (j))Pi (l)(j=1)(z-mu (j)) where \mu (j)\ < 1(j = 1,...,l), \rho (j)\ > 1(j = 1,...,h), c(0) constant; nu (1),...,nu (p) an ordered set of complex numbers of unequal moduli - ordered by increasing moduli so that that phi (z) not equal 0 for \z\ = 1, and q = #{nu (j) : \nu (j)\ < 1} greater than or equal to l. Then T-phi is Fredholm and index T-phi = i = -(q - l). phi has a decomposition phi = Theta (1)<(<Theta>)over bar>(2). f where f is outer and Theta (1) and Theta (2) are Blaschke products where the zeros of Theta (1) are the zeros of phi in the disk and the zeros of Theta (2) are the poles of phi inside the disk. Decompose Theta (1) into a product of Blaschke factors Theta (1) = Theta (max).Theta (min) where Theta (max) is obtained from Theta (1) by picking the largest tin terms of modulus) q - l zeros of Theta (1) inside the disk while Theta (min) has the remaining zeros inside the disk. Let S be the unilateral shift on the Hardy space H-2(T). Let P-n be the orthogonal projection to ker S*(n+1), and let P(T-<(<phi>)over bar>) be the projection to kernel T-<(<phi>)over bar>. For the injective Toeplitz operator T-phi we prove lim(n --> infinity) \det(PnTphi \kerS*n+1)\/det(P(T-<(<phi>)over bar>)S-\kerT<(<phi>)over bar>(n+1))\ /expn+1/2 pi integral (2 pi)(0)log\phi (e(i theta))\d theta = det(1-(P(T-<(<Theta>)over bar>1)-P(T-<(<Theta>)over bar>2))(2))(\ran(1-(P(T<(<Theta>)over bar>1)-(P(T<(<theta>)over bar>2))2))(1/2) x Pi (\x\<1)\c(x)(f, Theta (2)/Theta (min))\, where c(x)(h(1), h(2)) denotes the tame symbol of h(1), h(2) at the point x; the limit is computed explicitly in terms of the zeros and poles of phi. This is achieved using the theory of perturbation vectors, sigma (T1,T2) introduced in [2]. If phi = Omega (1)<(<Omega>)over bar>(2), for finite Blaschke products Omega (1), Omega (2) of degrees q and l respectively with q greater than or equal to l, fix an orthonormal basis B of ker T-<(<Omega>)over bar>1. Then lim(n --> infinity)\det(PnTphi \kerS*n+1)\/det(P(T-<(<phi>)over bar>)S-\kerT<(<phi>)over bar>(n+1))\ = parallel to sigma (1),T(phi)T(phi -1)parallel to = det((TphiTphi -1)(\rangeT phi))=Sigma (GammaB)cos(2)(Lambda (l)span Gamma (B), Lambda K-l(Omega2)). Here K-Omega2 = H(2)circle minus H-2(2); and the sum is extended over all the ((q)(l)) subsets Gamma (B) subset of B of cardinality l. Additionally, if l = q, or q = l + 1, parallel to sigma (1),T(phi)T(phi -1)parallel to is computed in terms of hyperbolic angles of parallelism.
