A Subrecursive Refinement of the Fundamental Theorem of Algebra

Abstract

Let us call an approximator of a complex number α any sequence γ ₀,γ ₁,γ ₂,... of rational complex numbers such that

$$|\gamma_t-\alpha|\le \frac{1}{t+1},\ \ t=0,1,2,\ldots$$

Denoting by ℕ the set of the natural numbers, we shall call a representation of α any 6-tuple of functions f ₁,f ₂,f ₃,f ₄,f ₅,f ₆ from ℕ into ℕ such that the sequence γ ₀,γ ₁,γ ₂,... defined by

$$\gamma_t=\frac{f_1(t)-f_2(t)}{f_3(t)+1}+\frac{f_4(t)-f_5(t)}{f_6(t)+1}i,\ \ t=0,1,2,\ldots\,,$$

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 α ₀,α ₁,...,α _{N − − 1} of N complex numbers into the components of some representation of some root of the corresponding polynomial P(z)=z ^N+α _{N − − 1} z ^{N − − 1}+⋯+α ₁ z+α ₀.