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

[Timothy Y. Chow]

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.

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Tim Chow