Witt vectors and Witt polynomials and the representation of p-adic numbers as space-time surfaces

We have had very inspiring discussions with Robert Paster, who advocates the importance of universal Witt vectors and Witt polynomials (see this) in the modelling of the brain, have been very inspiring. As the special case Witt vectors code for p-adic number fields. Witt polynomials are characterized by their roots, and the TGD videw about space-time surfaces both as generalized numbers and representations of ordinary numbers, inspires the idea how the roots of Witt polynomials could be represented as space-time surfaces in the TGD framework. This would give a representation of p-adic numbers as space-time surfaces.

Could the prime polynomial pairs (g₁,g₂): C²→ C² and (f₁,f₂): H=M⁴× CP₂→ C² (perpaps states of pure, non-reflective awareness) characterized by small primes give rise to p-adic numbers represented in terms of space-time surfaces such that these primes could correspond to p-adic primes?

1. Universal Witt vectors and polynomials can be assigned to any commutative ring R. Witt vectors X_n define sequences of elements of R and Universal Witt polynomials W_n(X_n) define a sequence of polynomials of order n. In the general case the Witt polynomial can be written as W_n=∑_d|n d X^{n/d, where d is a divisor of n, with 1 and n included. Clearly W_n characterizes the number theoretical anatomy of n.}
2. The roots of W_n characterize W_n it and just for fun one can ask whether also W_ns could determine space-time surfaces as a representation for the anatomy of integer n and n itself. The roots of W_n would define a set of disjoint surfaces. One can ask whether this kind of representation of polynomials in terms of their roots in turn represented in terms of space-time surfaces is a universal feature of mathematical cognition.
3. Cognition would really create worlds! In Finland we have Kalevala as a national epic and it roughly says that things were discovered by first discovering the word describing the thing. Something similar appears in the Bible: “In the beginning was the Word, and the Word was with God, and the Word was God. Word is world!

Also p-adic numbers could be represented in terms of space-time surfaces.

1. UWVs are defined for any Abelian ring, not only ordinary numbers. For instance, the function pairs (f₁,f₂): M⁴→ C² define space-time surfaces as their roots form an Abelian ring with respect to element-wise sum and multiplication. One could therefore consider the n:th powers of (f₁ⁿ,f₂ⁿ) of (f₁,f₂). The roots of the polynomial W_n contain (f₁,f₂)=0 but the remaining roots of W_n are different and correspond to pairs of algebraic numbers (f₁,f₂)=(r₁,r₂). W_n gives rise to n disjoint space-time surfaces as its roots.
2. The function pairs g= (g₁,g₂): C²→ C² define by the iteration reflective hierarchies thoughts about… about prime thoughts defined by prime pairs (f₁,f₂), which do not allow further functional composition to gcirc f Also the pairs g allow prime pairs and their algebra Witt vectors and corresponding Witt polynomials. The roots for (g₁,g₂), which are functional primes and its powers (g₁ⁿ,g₂ⁿ) correspond to discrete points.
3. Could the Witt polynomials relate to the abstraction hierarchy in which the degrees of polynomials appear as powers of primes associated with the prime function pair (f₁,f₂) or (g₁,g₂). For a general Witt polynomial W_n, the degree increases very slowly with n rather than exponentially. Special conditions on the values of n are required.

However, if one restricts the consideration to p-adic numbers, the values of n come as powers p^k, in the same way as in the case of functional iteration, and a good candidate for p could be the prime associated with a prime polynomial g or prime polynomial f.

However, in the general case 3 primes characterize the pair f as a prime and 2 primes characterize the prime pair g as prime with respect to functional composition. Does this mean multi-p p-adicity so that the multi-p-adic integers would be a power series of an integer n divisible by several primes and belong to the intersection of several p-adic number fields. The elements of the p-adic number field would be represented as p-adic Witt vectors represented in terms of Witt polynomials represented in terms of their roots. Thoughts as elements of p-adic number fields would represent unions of space-time surfaces! It seems that the small primes could indeed define p-adic primes and p-adic topologies. As a matter of fact, the roots of W_n are disjoint conforms with the idea about total disconnectedness of the p-adic topology in which all sets are both open and closed. The roots of W_n define discriminant D, decomposing to the powers of ramified primes identified as large p-adic primes in p-adic mass calculations (see this).

Could this give a generalization of the p-adic length scale hypothesis p∼ q^k, such that q is a small prime assignable to pair (f₁,f₂), which is prime with respect to functional composition, and p is the ramified prime of W_q^k. Note that the spectrum of ramified primes would depend on the power q^k. One should be able to show that the spectrum contains p∼ 2^k. Large ramified primes are indeed possible since the degree of W_q^k increases exponentially with k and therefore also the number of roots as its degree q^k. For electrons one would have p=M₁₂₇ = 2¹²⁷-1 and k=127. The coefficients of W_q^k are powers qⁱ, i

See the article A more detailed view about the TGD counterpart of Langlands correspondence or the chapter About Langlands correspondence in the TGD framework.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.