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

We have had very inspiring discussions with Robert Paster, who advocates the importance of universal Witt Vectors (UWVs) 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 view about space-time surfaces both as generalized numbers and representations of ordinary numbers, inspires the idea how the roots of for suitably identified Witt polynomials could be represented as space-time surfaces in the TGD framework. This would give a representation of generalized 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 ordinary p-adic primes? Same question applies to the pairs (f₁,f₂) which are functional primes.

1. Universal Witt vectors and polynomials can be assigned to any commutative ring R, not only p-adic integers. Witt vectors X_n define sequences of elements of a ring R and Universal Witt polynomials W_n(X₁,X₂,…,X_n) define a sequence of polynomials of order n. In the case of p-adic number field X_n correspond to the pinary digit of power pⁿ and can be regarded as elements of finite field F_p which can be also mapped to phase factors exp(ik 2π/p). The motivation for Witt polynomials is that the multiplication and sum of p-adic numbers can be done in a component-wise manner for Witt polynomials whereas for pinary digits sum and product affect the higher pinary digits in the sum and product.

In the general case, the Witt polynomial as a polynomial of several variables can be written as W_n(X₀,X₁,…)=sum_{dmid n} d X_d^{n/d, where d is a divisor of n, with 1 and n included.}

The function pairs (f₁,f₂): M⁴→ C² define a ring-like structure. Product and sum are well-defined for these pairs. The function pair related to (f₁,f₂) by a multiplication by a function pair (h₁,h₂), which vanishes nowhere in CD, defines the same space-time surface as the original one is equivalent with the original one. Note that also the powers (f₁ⁿ,f₂ⁿ) define the same 4-surfaces as (f₁,f₂).

The degrees for the product of polynomial pairs (P₁,P₂) and (Q₁,Q₂) are additive. In the sum, the degree of the sum is not larger than the larger degree and it can happen that the highest powers sum up to zero so that the degree is smaller. This reminds of the properties of non-Archimedean norm for the p-adic numbers. The zero element defines the entire H as a root and the unit element does not define any space-time surface as a root.

For the pairs (g₁,g₂) also functional composition is possible and the degrees are multiplicative in this operation.

Functional primes (f₁,f₂) define analogs of ordinary primes and the polynomials with degrees associated with the 3 complex coordinates of H below the primes associated with these coordinates are analogous to pinary digits. Also the pairs (g₁,g₂) define functional primes both with respect to powers defined by element-wise product and functional composition.

Generalization of Witt polynomials

Could a representation of polynomials, in particular the analogs of Witt polynomials in terms of their roots in turn represented in terms of space-time surfaces, be a universal feature of mathematical cognition? If so, 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!

Could p-adic numbers or their generalization for functional primes (f₁,f₂) have a representation in terms of Witt polynomials coded by their roots defining space-time surfaces.

1. W_n is a polynomial of n arguments X_k whereas the arguments of the polynomials defining space-time surfaces correspond to 3 complex H coordinates. In the p-adic case the factors d are powers of p. X_d are analogous to elements of a finite field as coefficients of powers of p.
2. There are two cases to consider. The Witt polynomials assignable to the space-time surfaces (f₁,f₂)=(0,0): H→ C² using element-wise sum and product. For the pairs g=(g₁,g₂)=(0,0): C²→ C² one can consider sum and element-wise product giving gⁿ= (g₁ⁿ,g₂ⁿ) and the sum or functional composition giving g(g(…g)…). The latter option looks especially attractive. One reason is that by the previous considerations the prime surface pairs (f₁,f₂) might be two simple. For instance the iterations (g₁,g₂) with prime degree 2,3,.. could give a justification for the p-adic length scale hypothesis and its generalization.

Consider first the pairs (f₁,f₂): H→ C².

1. If the space-time surface (f₁,f₂)=(0,0) is prime with respect to the functional composition f→ g(f), it naturally generalizes the p-adic prime p so that one would have p^k→ (f₁,f₂)^k and n₁=n₂.

X_k are the analogs of pinary digits as elements of finite fields. Could they correspond to polynomials with the 3 degrees smaller than the corresponding prime degree assignable to the prime polynomial (f₁,f₂)?

With these identifications it might be possible to generalize the Witt polynomials to their functional variants as such and find its roots represented as space-time surfaces. These surfaces would represent the functional analog of the p-adic number field. One can also assign to the functional p-adic numbers ramified primes defining ordinary p-adic primes. Each functional p-adic number would define ramified primes and these would correspond to the p-adic primes.

f_i are labelled by 3 ordinary primes p_r(f_i), r=1,2,3, rather than single prime p and by the earlier argument one can restrict the condition to f₁.

Every functional p-adic number corresponds to its own ramified primes determined by the roots of its Witt polynomial. There is a huge number of these generalized p-adic numbers. Could some special functional p-adic primes correspond to elementary particles? The simplest generalized p-adic number corresponds to a functional prime and in this case the surface in question would correspond to (f₁,f₂)=(0,0) (could this be interpreted as stating the analog of mod ~p=0 condition). These prime surfaces might be too simple and it is not easy to understand how the large values of p–adic primes could be understood. One can ask whether the analogs of ramified primes for the Witten polynomials assignable abstraction hierarchies g(g(…(f)…) and powers gⁿ=(g₁ⁿ,g₂ⁿ) for which the degree of the polynomials is n×p, p the prime assignable to g.

1. The ramified primes for the Witten polynomials for g(g(…(f)…) and gⁿ defining analogs of powers pⁿ of p-adic numbers. Note that the roots of g(g(…(f)…) are a property of g(g(…(g)…) and do not depend on f in case that they exist as surfaces inside the CD.
2. The interesting question is whether and how the ramified primes could relate to the ramified primes assignable to a generalized Witt polynomial W_n. The iterated action of prime g giving g(g(…(f)…) is the best candidate. There is hope that even the p-adic length scale hypothesis could be understood as a ramified primes assignable to some functional prime. The large values of p-adic primes require that very large ramified primes for the functional primes (f₁,f₂). This would suggest that the iterate g(g….g(f)…) acting on prime f is involved. For p∼ q^k, k^th power of g characterized by prime g is the first guess.

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.

Source: https://matpitka.blogspot.com/2025/03/witt-vectors-and-polynomials-and.html