A Subrecursive Refinement of the Fundamental Theorem of Algebra

Let us call an approximator of a complex number α any sequence γ0, γ1, γ2, . . . of rational complex numbers such that |γt − α| ≤ 1 t+ 1 , t = 0, 1, 2, . . . Denoting by N the set of the natural numbers, we shall call a representation of α any 6-tuple of functions f1, f2, f3, f4, f5, f6 from N into N such that the sequence γ0, γ1, γ2, . . . defined by γt = f1(t)− f2(t) f3(t) + 1 + f4(t)− f5(t) f6(t) + 1 i, t = 0, 1, 2, . . . , is an approximator of α. For any representations of the members of a finite sequence of complex numbers, the concatenation of these representations will be called a representation of the sequence in question (thus the representations of the sequence have a length equal to 6 times the length of the sequence itself). By adapting a proof given by P. C. Rosenbloom we prove the following refinement of the fundamental theorem of algebra: for any positive integer N there is a 6-tuple of computable operators belonging to the second Grzegorczyk class and transforming any representation of any sequence α0, α1, . . . , αN−1 of N complex numbers into the components of some representation of some root of the corresponding polynomial P (z) = z + αN−1z + · · ·+ α1z + α0.