Does the Notion of Polynomial of Infinite Order Make Sense?

Abstract

M⁸-H duality relates number theoretical and geometric visions of physics in the TGD framework. At the level of M⁸ polynomials with rational coefficients would define the space-time surfaces as the "roots" of their complexified octonionic continuations. The basic dynamical principle states that they have associative normal spaces. In principle, also an analytic function with rational Taylor coefficients is possible and can give rise to transcendental extensions. A longstanding question has been whether it makes sense to talk about polynomials of infinite degree. It turns out that if the polynomials of infinite degree exist, they must correspond to composites for an infinite number of polynomials. This follows from the fact that both finite and infinite Galois groups must be profinite so that an infinite Galois group is a Galois group of ...extensions of extensions.....of  rationals. The example in which the polynomials of form P = P ∘ R where Q is an infinite composite of a single polynomial Q vanishing at origin and having it as a critical point has as a basin of attraction a set having Julia set as boundary. All points in the basin of attraction for origin are roots at the limit. All points in the basin of attraction for origin are roots at the limit so that an algebraic completion of rationals to complex numbers would result.