# A Subrecursive Refinement of the Fundamental Theorem of Algebra

• Peter Peshev
• Dimiter Skordev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)

## 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.

## Keywords

Fundamental theorem of algebra Rosenbloom’s proof computable analysis computable operator second Grzegorczyk class

## References

1. 1.
Grzegorczyk, A.: Some Classes of Recursive Functions, Dissertationes Math. (Rozprawy Mat.), vol. 4, Warsaw (1953)Google Scholar
2. 2.
Grzegorczyk, A.: Computable functionals. Fund. Math. 42, 168–202 (1955)
3. 3.
Peshev, P.: A subrecursive refinement of the fundamental theorem of algebra. Sofia University, Sofia (master thesis, in Bulgarian) (2005)Google Scholar
4. 4.
Rosenbloom, P.C.: An elementary constructive proof of the fundamental theorem of algebra. The Amer. Math. Monthly 52(10), 562–570 (1945)
5. 5.
Skordev, D.: Computability of real numbers by using a given class of functions in the set of the natural numbers. Math. Logic Quarterly 48(suppl. 1), 91–106 (2002)
6. 6.
Weihrauch, K.: Computable Analysis. In: An Introduction. Springer, Heidelberg (2000)