Sunday, July 14, 2019

What extensions of rationals could be winners in the fight for survival?

It would seem that the fight for survival is between extensions of rationals rather than individual primes p. Intuition suggests that survivors tend to have maximal number of ramified primes. These number theoretical speciei can live in the same extension - to "co-operate".

Before starting one must clarify for myself some basic facts about extensions of rationals.

  1. Extension of rationals are defined by an irreducible polynomial with rational coefficients. The roots give n algebraic
    numbers which can be used as a basis to generate the numbers of extension ast their rational linear combinations. Any number of extension can be expressed as a root of an irreducible polynomial. Physically it is is of interest, that in octonionic picture infinite number of octonionic polynomials gives rise to space-time surface corresponding to the same extension of rationals.

  2. One can define the notion of integer for extension. A precise definition identifies the integers as ideals. Any integer of extension are defined as a root of a monic polynomials P(x)=xn+pn-1xn-1+...+p0 with integer coefficients. In octonionic monic polynomials are subset of octonionic polynomials and it is not clear whether these polynomials could be all that is needed.

  3. By definition ramified primes divide the discriminant D of the extension defined as the product D=∏i≠ j (ri-rj) of differences of the roots of (irreducible) monic polynomial with integer coefficients defining the basis for the integers of extension. Discriminant has a geometric interpretation as volume squared for the fundamental domain of the lattice of integers of the extension so that at criticality this volume interpreted as p-adic number would become small for ramified primes an vanish in O(p) approximation. The extension is defined by a polynomial with rational coefficients and integers of extension are defined by monic polynomials with roots in the extension: this is not of course true for all monic polynomials polynomial (see this).

  4. The scaling of the n-1-tuple of coefficients (pn-1,.....,p1) to (apn-1,a2pn-1.....,anp0) scales the roots by a: xn→ axn. If a is rational, the extension of rationals is not affected. In the case of monic polynomials this is true for integers k. This gives rational multiples of given root.

    One can decompose the parameter space for monic polynomials to subsets invariant under scalings by rational k≠ 0. Given subset can be labelled by a subset with vanishing coefficients {pik}. One can get rid of this degeneracy by fixing the first non-vanishing pn-k to a non-vanishing value, say 1. When the first non-vanishing pk differs from p0, integers label the polynomials giving rise to roots in the same extension. If only p0 is non-vanishing, only the scaling by powers kn give rise to new polynomials and the number of polynomials giving rise to same extension is smaller than in other cases.

    Remark: For octonionic polynomials the scaling symmetry changes the space-time surface so that for generic polynomials the number of space-time surfaces giving rise to fixed extension is larger than for the special kind polynomials.

Could one gain some understanding about ramified primes by starting from quantum criticality? The following argument is poor man's argument and I can only hope that my modest technical understanding of number theory does not spoil it.
  1. The basic idea is that for ramified primes the minimal monic polynomial with integer coefficients defining the basis for the integers of extension has multiple roots in O(p)=0 approximation, when p is ramified prime dividing the discriminant of the monic polynomial. Multiple roots in O(p)=0 approximation occur also for the irreducible polynomial defining the extension of rationals. This would correspond approximate quantum criticality in some p-adic sectors of adelic physics.

  2. When 2 roots for an irreducible rational polynomial co-incide, the criticality is exact: this is true for polynomials of rationals, reals, and all p-adic number fields. One could use this property to construct polynomials with given primes as ramified primes. Assume that the extension allows an irreducible olynomial having decomposition into a product of monomials =x-ri associated with roots and two roots r1 and r2 are identical: r1=r2 so that irreducibility is lost.

    The deformation of the degenerate roots of an irreducible polynomial giving rise to the extension of rationals in an analogous manner gives rise to a degeneracy in O(p)=0 approximation. The degenerate root r1=r2 can be scaled in such a manner that the deformation r2=r1(1+q)), q=m/n=O(p) is small also in real sense by selecting n>>m.

    If the polynomial with rational coefficients gives rise to degenerate roots, same must happen also for monic polynomials. Deform the monic polynomial by changing (r1,r2=r1) to (r1,r1(1+r)), where integer r has decomposition r=∏piki to powers of prime. In O(p)=0 approximation the roots r1 and r2 of the monic polynomial are still degenerate so that pi represent ramified primes.

    If the number of pi is large, one has high degree of ramification perhaps favored by p-adic evolution as increase of number theoretic co-operation. On the other hand, large p-adic primes are expected to correspond to high evolutionary level. Is there a competition between large ramified primes and number of ramified primes? Large heff/h0=n in turn favors large dimension n for extension.

  3. The condition that two roots of a polynomial co-incide means that both polynomial P(x) and its derivative dP/dx vanish at the roots. Polynomial P(x)= xn +pn-1xn-1+..p0 is parameterized by the coefficients which are rationals (integers) for irreducible (monic) polynomials. n-1-tuple of coefficients (pn-1,.....,p0) defines parameter space for the polynomials. The criticality condition holds true at integer points n-1-D surface of this parameter space analogous to cognitive representation.

    The condition that critical points correspond to rational (integer) values of parameters gives an additional condition selecting from the boundary a discrete set of points allowing ramification. Therefore there are strong conditions on the occurrence of ramification and only very special monic polynomials are selected.

    This suggests octonionic polynomials with rational or even integer coefficients, define strongly critical surfaces, whose p-adic deformations define p-adically critical surfaces defining an extension with ramified primes p. The condition that the number of rational critical points is non-vanishing or even large could be one prerequisite for number theoretical fitness.

  4. There is a connection to catastrophe theory, where criticality defines the boundary of the region of the parameter space in which discontinuous catastrophic change can take place as replacement of roots of P(x) with different root. Catastrophe theory involves polynomials P(x) and their roots as well as criticality. Cusp catastrophe is the simplest non-trivial example of catastrophe surface with P(x)= x4/4-ax-bx2/2: in the interior of V-shaped curve in (a,b)-plane there are 3 roots to dP(x)=0, at the curve 2 solutions, and outside it 1 solution. Note that now the parameterization is different from that proposed above. The reason is that in catastrophe theory diffeo-invariance is the basic motivation whereas in M8 there are highly unique octonionic preferred coordinates.

If p-adic length scale hypothesis holds true, primes near powers of 2, prime powers, in particular Mersenne primes should be ramified primes. Unfortunately, this picture does not allow to say anything about why ramified primes near power of 2 could be interesting. Could the appearance of ramified primes somehow relate to a mechanism in which p=2 as a ramified prime would precede other primes in the evolution. p=2 is indeed exceptional prime and also defines the smallest p-adic length scale.

For instance, could one have two roots a and a+2k near to each other 2-adically and could the deformation be small in the sense that it replaces 2k with a product of primes near powers of 2: 2k = ∏i 2ki→ ∏ipi, pi near 2ki? For the irreducible polynomial defining the extension of rationals, the deforming could be defined by a→ a+2k could be replaced by a→ a+2k/N such that 2k/N is small also in real sense.

See the article Trying to understand why ramified primes are so special physically or the chapter TGD View about Coupling Constant Evolution.

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

Articles and other material related to TGD.

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