Finding the roots of polynomials defined by infinite primes

Infinite primes identifiable as analogs of free single particle states and bound many-particle states of a repeatedly second quantized supersymmetric arithmetic quantum field theory correspond at n:th level of the hierarchy to irreducible polynomials in the variable Xn which corresponds to the product of all primes at the previous level of hierarchy. At the first level of hierarchy the roots of this polynomial are ordinary algebraic numbers but at higher levels they correspond to infinite algebraic numbers which are somewhat weird looking creatures. These numbers however exist p-adically for all primes at the previous levels because one one can develop the roots of the polynomial in question as powers series in Xn-1 and this series converges p-adically. This of course requires that infinite-p p-adicity makes sense. Note that all higher terms in series are p-adically infinitesimal at higher levels of the hierarchy. Roots are also infinitesimal in the scale defined Xn. Power series expansion allows to construct the roots explicitly at given level of the hierarchy as the following induction argument demonstrates.

  1. At the first level of the hierarchy the roots of the polynomial of X1 are ordinary algebraic numbers and irreducible polynomials correspond to infinite primes. Induction hypothesis states that the roots can be solved at n:th level of the hierarchy.

  2. At n+1:th level of the hierarchy infinite primes correspond to irreducible polynomials

    Pm(Xn+1)= ∑s=0,...,m ps Xsn+1 .

    The roots R are given by the condition

    Pm(R)=0 .

    The ansatz for a given root R of the polynomial is as a Taylor series in Xn:

    R= ∑ rkXnk ,

    which indeed converges p-adically for all primes of the previous level. Note that R is infinitesimal at n+1:th level. This gives

    Pm(R)=∑s=0,...,m ps (∑ rkXnk)s=0 .

    1. The polynomial contains constant term (zeroth power of Xn+1 given by

      Pm(r0)=∑s=0,...,m pr r0s .

      The vanishing of this term determines the value of r0. Although r0 is infinite number the condition makes sense by induction hypothesis. One can indeed interpret the vanishing condition Pm(r0)=0 as a vanishing of a polynomial at the n:th level of hierarchy having coefficients at n-1:th level and continue the process down to the lowest level of hierarchy obtaining m:th order polynomial at each step. At the lowest level of the hierarchy one obtains just ordinary polynomial equation having finite algebraic numbers as roots.

    2. If one has found the values of r0 one can solve the coefficients rs, s>0 as linear expressions of the coefficients rt, t0.

    3. The naive expectation is that the fundamental theorem of algebra generalizes so that that the number of different roots r0 would be equal to m in the irreducible case. This seems to be the case. Suppose that one has constructed a root R of Pm. One can write Pm(Xn+1) in the form

      Pm(Xn+1)= (Xn+1-R) × Pm-1(Xn+1) ,

      and solve Pm-1 by expanding Pm as Taylor polynomial with respect to Xn+1-R. This is achieved by calculating the derivatives of both sides with respect to Xn+1. The derivatives are completely well-defined since purely algebraic operations are in question. For instance, at the first step one obtains Pm-1(R)=(dPm/dXn+1)(R). The process stops at m:th step so that m roots are obtained.

What is remarkable that the construction of the roots at the first level of the hierarchy forces the introduction of p-adic number fields and that at higher levels also infinite-p p-adic number fields must be introduced. Therefore infinite primes provide a higher level concept implying real and p-adic number fields. If one allows all levels of the hierarchy, a new number Xn must be introduced at each level of the hierarchy. About this number one knows all of its lower level p-adic norms and infinite real norm but cannot say anything more about them. The conjectured correspondence of real units built as ratios of infinite integers and zero energy states however means that these infinite primes would be represented as building blocks of quantum states and that the points of imbedding space would have infinitely complex number theoretical anatomy able to represent zero energy states and perhaps even the world of classical worlds associated with a given causal diamond.

For background see the chapter TGD as a Generalized Number Theory III: Infinite Primes and for the pdf version of the argument the chapter Non-Standard Numbers and TGD.