Math Music and Mathematics; From Pythagoras to Fractals

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N. Consider for k = 0, . . , n − 2 the pair of second-order recursions in F[z]: uk+2 (z) = uk (z) − qk+1 (z)uk+1 (z), vk+2 (z) = vk (z) − qk+1 (z)vk+1 (z), with initial conditions u0 (z) = 1, u1 (z) = 0, v0 (z) = 0, v1 (z) = 1. By the Euclidean algorithm, rk+2 = rk − qk+1 rk+1 for k = 0, . . n − 1. Thus sk := uk a + vk b satisfies the same recursion sk+2 = sk − qk+1 sk+1 with s0 = a = r0 , s1 = b = r1 . This implies rk = sk for k = 0, . . , n − 1. We conclude that un (z), vn (z) is the solution to the Diophantine equation rn = un a + vn b, where rn is a greatest common divisor of a, b.

A principal ideal domain (PID) is an integral domain such that every ideal of the ring is a principal ideal. Standard examples of PIDs are the ring of integers and the ring of polynomials in a single variable over a field. Since the additive group (R, +) of a ring is abelian, every ideal I of R is a normal subgroup of (R, +). Therefore, the factor group of coset elements R/I := {x + I | x ∈ R} is a group. Here we use the notation x + I := {x + a | a ∈ I}. The elements x + I of R/I are called residue classes (or cosets) of x modulo I.

2 Divisibility and Coprimeness of Polynomials 33 Notice that the first block of columns of Res(p, q) has n columns while the second block has m columns. 9. Two polynomials p(z), q(z) ∈ F[z] are coprime if and only if the Sylvester resultant Res(p, q) is invertible. Proof. Let F[z]

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