FOM: Enormous Integers/AlgGeom

[Harvey Friedman]

In 40:Enomrous Integers in Algebraic Geometry, I wrote:

>J(C,4,4) is the longest possible length starting with a complex algebraic
>set in 4 dimensions of degree 4, and successively taking proper algebraic
>subsets of exactly one higher degree. The result will still hold for the
>complex numbers instead of R.
>
>THEOREM 5. J(R,4,4) is greater than 2^2^2^2^2^2 = 2^2^2^16 = 2^2^65,536.

Of course, I meant J(C,4,4) there instead of J(R,4,4). The reason I wrote
J(R,4,4) was that I was trying to do this for the reals, but had not seen
my way thru it.

I can now tell you that I have been able to do this for the reals, with the
same lower bound. In fact, all the results cited there hold for the reals.

In fact, more is true. I can do it for the rationals, with the same
results. It is clear that J(Q,k,n) is at most J(F,k,n) for any field F of
characteristic zero, so my lower bounds for Q imply those for R and C.

Also, J(C,4,5) >= J(R,4,5) >= J(Q,4,5) >= 2^2^2^65,536. In 4 dimensions,
these lower bounds go up by an exponential every time you raise the
starting degree (in this case 5).

In 5 dimensions, the lower bounds are incomparably higher (the next level
of the Ackerman hierarchy after superexponentiation). I'll report on what
happens starting with a quadratic next time, plus more. I.e., J(Q,5,2).