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 X_n 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 X_n-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 X_n. 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

   P_m(X_n+1)= ∑_s=0,...,m p_s X^s_n+1 .

   The roots R are given by the condition

   P_m(R)=0 .

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

   R= ∑ r_kX_n^k ,

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

   P_m(R)=∑_s=0,...,m p_s (∑ r_kX_n^k)^s=0 .

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

      P_m(r₀)=∑_s=0,...,m p_r r₀^s .

      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

      P_m×m1(r₀)=0

      as a vanishing of a polynomial at the n:th level of hierarchy having coefficients at n-1:th level. Here m₁ is determined by the dependence on infinite primes of lower level expressible in terms of rational functions. One can continue the process down to the lowest level of hierarchy obtaining m×m₁...×m_k:th order polynomial at k:th step. At the lowest level of the hierarchy one obtains just ordinary polynomial equation having ordinary algebraic numbers as roots.

      One can expand the infinite primes as a Taylor expansion in variables X_i and the resulting number differs from an ordinary algebraic number by an infinitesimal in the multi-P infinite-P p-adic topology defined by any choice of n-plet of infinite-P p-adic primes (P₁,...,P_n) from subsequent levels of the hierarchy appearing in the expansion. In this sense the resulting number is infinitely near to an ordinary algebraic number and the structure is analogous to a completion of algebraic numbers to reals. Could one regard this structure as a possible alternative view about reals remains an open question. If so, then also reals could be said to have number theoretic anatomy.

   2. If one has found the values of r₀ one can solve the coefficients r_s, s>0 as linear expressions of the coefficients r_t, t<s and thus in terms of r₀.

   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 P_m. One can write P_m(X_n+1) in the form

      P_m(X_n+1)= (X_n+1-R) × P_m-1(X_n+1) ,

      and solve P_m-1 by expanding P_m as Taylor polynomial with respect to X_n+1-R. This is achieved by calculating the derivatives of both sides with respect to X_n+1. The derivatives are completely well-defined since purely algebraic operations are in question. For instance, at the first step one obtains P_m-1(R)=(dP_m/dX_n+1)(R). The process stops at m:th step so that m roots are obtained. At lower levels similar branching occurs just as it occurs for polynomials of several variables.

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 X_n 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 of "Physics as a Generalized Number Theory".