A Subrecursive Refinement of the Fundamental Theorem of Algebra

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


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.


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)MathSciNetzbMATHGoogle Scholar
  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)MathSciNetCrossRefzbMATHGoogle Scholar
  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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Weihrauch, K.: Computable Analysis. In: An Introduction. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Peter Peshev
    • 1
  • Dimiter Skordev
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceUniversity of SofiaSofiaBulgaria

Personalised recommendations