[FOM] Koenigsmann's universal definition of Z in Q

[Mario Carneiro]

Perhaps this is a silly question, but... The theorem starts with a pair of
existentials, so in principle we should have an *actual* concrete
polynomial to look at here. Can the result be stated for any particular
polynomial? Do we have any bounds on n or the height of g? Or is the method
inherently nonconstructive?

Mario

On Fri, Jan 22, 2016 at 4:37 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

> This is old news but I only learned about it today, and since it doesn't
> seem to have been mentioned on FOM, I figured that some subscribers may not
> have heard of it either.
>
> Jochen Koenigsman has proved:
>
> Theorem.  For some positive integer n, there exists a polynomial
> g in Z[t; x_1, ..., x_n] such that, for any t in Q,
>
>    t is in Z  iff  forall x_1,...,x_n in Q: g(t; x_1, ..., x_n) != 0.
>
> In other words, Z is universally definable in Q.  This does not quite
> resolve Hilbert's tenth problem over Q---for that, one wants to show that Z
> is *existentially* definable in Q---but it comes close.
>
> Koenigsmann posted a preprint to the ArXiv back in 2010.  I found out
> about it only today because I happened to see the published version:
>
> Jochen Koenigsmann, Defining Z in Q, Ann. Math. 183 (2016), 73-93.
>
> I quote a few paragraphs from Koenigsmann's paper:
>
> ---
>
> Like all previous definitions of Z in Q, we use elementary facts on
> quadratic forms over R and Q_p, together with the Hasse-Minkowski
> local-global principle for quadratic forms.  What is new in our approach is
> the use of the Quadratic Reciprocity Law (e.g., in Propositions 10 or 16)
> and, inspired by the model theory of local fields, the transformation of
> some existential formulas into universal formulas (Step 4).  A technical
> key trick is the existential definition of the Jacobson radical of certain
> rings (Step 3) that makes implicit use of so-called `rigid elements' as
> they occur, e.g., in [Koe95].
>
> Step 1: Diophantine definition of quaternionic semi-local rings a la
> Poonen.  The first step modifies Poonen's proof ([Poo09a]), thus arriving
> at a formula for Z in Q that, like the formula in his Theorem 4.1, has two
> foralls followed by seven thereexists, but we managed to bring down the
> degree of the polynomial involved from 9244 to 8.
>
> Step 2: Towards a uniform diophantine definition of all Z_(p)'s in Q.  We
> will present a diophantine definition for the local rings Z_(p) = Z_p n Q
> depending on the congruence of the prime p modulo 8 and involving p (and if
> p=1 mod 8 an auxiliary prime q) as a parameter.
>
> Step 3: An existential definition for the Jacobson radical.  We will show
> that, for some rings R occurring in Proposition 10, the Jacobson radical
> J(R) can be defined by an existential formula.  This will also give rise to
> new diophantine predicates in Q.
>
> Step 4: From existential to universal.
>
> ---
>
> Tim Chow
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