## 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 - At the first level of the hierarchy the roots of the polynomial of X
_{1}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. - 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 _{k}X_{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_{k}X_{n}^{k})^{s}=0 .- The polynomial contains constant term (zeroth power of X
_{n+1}given byP _{m}(r_{0})=∑_{s=0,...,m}p_{r}r_{0}^{s}.The vanishing of this term determines the value of r _{0}. Although r_{0}is infinite number the condition makes sense by induction hypothesis. One can indeed interpret the vanishing condition P_{m}(r_{0})=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. - If one has found the values of r
_{0}one can solve the coefficients r_{s}, s>0 as linear expressions of the coefficients r_{t}, t~~0.~~ ~~The naive expectation is that the fundamental theorem of algebra generalizes so that that the number of different roots r~~_{0}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 formP _{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.
- The polynomial contains constant term (zeroth power of 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. |