A Subrecursive Refinement of the Fundamental Theorem of Algebra
Let us call an approximator of a complex number α any sequence γ 0,γ 1,γ 2,... of rational complex numbers such that
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
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.
KeywordsFundamental theorem of algebra Rosenbloom’s proof computable analysis computable operator second Grzegorczyk class
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