A subrecursive refinement of the fundamental theorem of algebra

Abstract. 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 =0, 1, 2,... t +1 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) + i, t =0, 1, 2,..., f6(t)+1 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 N + αN−1z N−1 + ···+ α1z + α0