de ning the integers in the rationals - umd

Defining the integers in the rationalsNYWIMN Conference

May 2, 2008

Carol Wood

Department of Mathematics and Computer ScienceWesleyan University

CSW Defining Z in Q

Defining Z in Q: Outline

Introduction

Julia Robinson’s thesis

Back to Hilbert’s tenth

Z in Q 60 years later

Poonen’s definition of Z in Q

CSW Defining Z in Q

Introduction

Defining Z in Q

IntroductionUndecidabilityDefinabilityHilbert’s Tenth Problem





CSW Defining Z in Q

Introduction Undecidability

Godelian phenomena

Several results concerning undecidability of the first order theory of analgebraic structure M have been obtained via a link to Godel’slncompleteness Theorem.Godel showed that the first order theory of the natural numbers N isundecidable.For example, suppose we can show that (a copy of) N, including itsaddition and multiplication, is definable in M.A decision procedure for the theory of M would then relativize to adecision procedure for the first order theory of < N,+, · >. But Godelshowed this to be impossible.Thus no decision procedure exists for the theory of M.

CSW Defining Z in Q 1 / 21

Introduction Definability

Definability

What do we mean by “first order definable”? Logicians know; for others,examples can help.Start with the ring of integers Z.Let’s see that N is definable in Z.First, a way NOT to do it:x ∈ N↔ x = 0 ∨ x = 1 ∨ x = 2 ∨ . . .

No infinite disjunctions!

The first-order definition we give involves some basic number theory.Lagrange’s Theorem to the rescue:x ∈ N↔ ∃a∃b∃c∃d(x = a2 + b2 + c2 + d2)



Continuing, with a minor rewrite

x ∈ N↔ ∃a∃b∃c∃d(0 = x − (a2 + b2 + c2 + d2))

Note also that this definition is of a particularly nice form, involving onlyexistential quantifiers.Thus we have an equation in four variables such that an integer x is anatural number exactly when the equation has a solution in Z for thosefour variables.This says that N is Diophantine in Z.



Squares

As a second example, we define the set of squares in Z.x is a square ↔ ∃a(a2 = x)).Thus the set of squares is also Diophantine.

Examples of definable subsets of N:

I the set of numbers x which are not powers of 2:∃a∃b(x = (2a + 3)b)

I the set of all composite x∃a∃b(x = (a + 2)(b + 2))



Nonsquares

The set of nonsquares is definable in Z:x is a nonsquare ↔ ∀a(a2 6= x).This definition is not optimal, in that it is not Diophantine.


Introduction Hilbert’s Tenth Problem

H10Related to the question of definability is the matter of the complexity ofthe definition.In 1900 Hilbert asked whether one could decide whether a Diophantineequation over the integers has an integral solution.The first order assertion that a given equation has a solution is anexistential formula.Godel’s Incompleteness Theorem does not settle this question. It may bepossible to decide the existential part of an undecidable theory; anexample of this behavior would be interesting!

We will return to H10 later...



Defining Z in Q

Introduction

Julia Robinson’s thesisForms representing integers




CSW Defining Z in Q

Julia Robinson’s thesis Forms representing integers

Representing integers

Theorem (Julia Robinson, 1948). The ring of integers Z is definable in thefield of rationals Q.The proof uses facts about which rationals can be represented by certainquadratic forms.Recall that a quadratic form f = a1X

21 + a2X

22 + · · ·+ asX

2s with

coefficients in a field K is said to represent 0 in K just in case theequation f (X ) = 0 has a solution X ∈ K s ,X 6= 0.Underlying the ingredients of JR’s proof is a fundamental local-globalprinciple:Hasse-Minkowski Theorem. A quadratic form f with integer coefficientsrepresents zero in Q if and only if f represents 0 mod n for every positiveinteger n and f represents 0 in R.



We take r ∈ Q and write r = nd with d ≥ 1 and such that n and d have no

common divisors > 1.Using forms which can only represent r when d is not divisible by certainprimes, JR was able to pick out those r for which d must equal 1.

I f = X 2 + Y 2 − pZ 2, where p is a prime integer, p ≡ 3 mod 4.FACT. f represents 2 + pr2 in Q↔ 2 - d and p - d

I g = X 2 + qY 2 − pZ 2, where p and q are odd primes with p ≡ 1 mod4 and q not a square mod p.FACT. g represents 2 + pqr2 in Q↔ p - d and q - d .

The proof of both facts relies on the Hasse-Minkowski Theorem.



Picking out the integers

Let φ(r , y , z) = ∃a∃b∃c(2 + yzr2 = a2 + yb2 − zc2).

Think of φ(r , y , z) as saying that r can be represented in both waysdescribed in the previous FACTS.

Let θ(r) = (φ(0, y , z) ∧ ∀w(φ(w , y , z)→ φ(w + 1, y , z))→ φ(r , y , z)).

Think of θ(r) as saying that induction applies to r .

Claim: a rational r is an integer ↔ ∀y∀zθ(r).

By defining Z in Q, JR proved that the theory of the rationals isundecidable.This formula is complicated, involving several alternations of quantifiers,∀∃∀It was the best available until quite recently.



Defining Z in Q

Introduction





CSW Defining Z in Q


H10 was solved in the negative in the late 1960’s by Matijasevich, buildingon previous work of Martin Davis, Hilary Putnam and Julia Robinson.Theorem (DPRM). There is no algorithm for deciding, given a polynomialf in n variables with integer coefficients, whether f = 0 has an integralsolution.An auxiliary question remains open: whether one can decide solvability forany third degree polynomial. Second degree: decidable. Fourth degree:undecidable.Note that we can formulate H10 for any ring R, simply by replacing“integer coefficients” by “coefficients in R” and “integral solution” by“solution in R”.



To transfer this negative result for Z to another ring, it is not enough tobe able to define Z in the ring, but it does suffice if one can define Z viaan existential formula.This has been achieved in some instances. H10 is a more subtle questionthan decidability of the full theory, and H10 remains open for severalundecidable rings, including Q and finite extensions of Q (i.e., numberfields).Also, there are no number fields for which it is known that H10 has apositive solution for its ring of integers.Many researchers have been involved in this fascinating enterprise, and Icannot list them all here, but will give references at the end.Ok, one name for this crowd: Alexandra Shlapentokh, PhD NYU 1988.



Defining Z in Q

Introduction



Z in Q 60 years laterEisentrager’s thesis


CSW Defining Z in Q

Z in Q 60 years later Eisentrager’s thesis

In her thesis (also at Berkeley), Kirsten Eisentrager worked on variousquestions around H10. She gave a new proof that, given a global field kand a nonarchimedean prime p, the set of elements of k integral at p isDiophantine.She did this in order to provide one of two key pieces needed to settle H10for positive characteristic cases, i.e., finite extensions of Fq(t).Of note for us today is her use of division algebras, specifically quaternionalgebras, which take the place of quadratic forms in earlier results alongthe same lines.


Z in Q 60 years later Eisentrager’s thesis

Quaternion algebras

Let K be a field of characteristic 6= 2, and let a, b ∈ K×.The quaternion algebra over K ,Ha,b = Ha,b(K ) is the K -algebra generatedby i and j , where i2 = a, j2 = b, and ij = k = −ji . Ha,b is four dimensionalover K , with basis {1, i , j , k}.most familiar case is the division algebra of real quaternions, with K = Rand a = b = −1.In general, Ha,b(K ) is either a division algebra or it is isomorphic toM2(K ), all two-by-two matrices over K .

We will be interested in Ha,b for K = Q and, for p prime, in K = Qp, thefield of p-adics .

For an element α = x1 + x2i + x3j + x4k , the reduced normn(α) = x2

1 − ax22 − bx2

3 + abx24 , and the reduced trace tr(α) = 2x1.



Defining Z in Q

Introduction




Poonen’s definition of Z in QQp and Zp

Another definability factRamified and unramified

A bit more detail

CSW Defining Z in Q

Poonen’s definition of Z in Q Qp and Zp

Crash course in the p-adics

We will think of the p-adics as follows (although much more could besaid):

Qp = {Σ∞i=kaipi |k ∈ Z, 0 ≤ ak ≤ p − 1}

Inside Qp there is a subring Zp of p-adic integers, namely all elementswith no non-zero coefficients for negative powers of p:

Zp = {Σ∞i=0aipi |0 ≤ ak ≤ p − 1}

We will refer to the reduction map from Zp to Fp given by

redp : Σ∞i=0akpk → a0

.


Poonen’s definition of Z in Q Another definability fact

Interlude: Zp is definable in Qp

For p 6= 2,Zp = {x |∃y(y2 = px2 + 1)}

To see this, think about what happens when you square a Laurent serieswith negative terms.


Poonen’s definition of Z in Q Ramified and unramified

Q to Qp

We consider Ha,b = Ha,b(Q); all four-dimensional central simple algebrasover Q arise in this form.Next we tensor Ha,b with Qp over Q. Here we are assuming p 6= 2; thecase p = 2 can be handled similarly.One of two things occurs:

I Ha,b ⊗Q Qp∼= Ha,b(Qp)

This is called the ramified case.

I Ha,b ⊗Q Qp∼= M2(Qp)

This is called the unramified, or split, case.



Overview of Poonen’s definitionTo define Z in Q, Poonen considers the set Sa,b(K ) of all possible reducedtraces of elements n Ha,b(K ) of reduced norm 1.

I When p does not ramify, all elements become traces, soSa,b(Qp) = Qp.

I If p ramifies, the situation is more complicated, and information canbe extracted when at least one of a and b is positive.



Final steps

In case a > 0 or b > 0, Hasse-Minkowski tells us that

Sa,b(Q) = Q ∩⋂

p Sa,b(Qp).

I The set of all rationals Ta,b of a certain form occurs as theintersection of the Zp’s for ramified p.

I As a, b range over the positive integers, the intersection of the Ta,b’sequals Z.


Poonen’s definition of Z in Q A bit more detail

Ingredients of proof

For q a pth power, let Uq = {s ∈ Fq|x2 − sx + 1 is irreducible over Fq}.I Uq 6= ∅, and for q > 11,Uq + Uq = Fq.

This requires a geometric argument, together with information aboutbounds on the numbers of points on curves on finite fields.

I If p is unramified, then red−1p (Up) ⊆ Sa,b(Qp) ⊆ Zp.

The proof uses the fact that s is a reduced trace of an element ofnorm 1 just in case x2 − sx + 1 is the reduced characteristicpolynomial of an element of Ha,b ⊗Qp.

I Let Ta,b = Q∩{s + s ′+ n|s, s ′ ∈ Sa,b, 0 ≤ n ≤ 2310 = 2 · 3 · 5 · 7 · 11}.Then Ta,b =

⋂Zp where the intersection is taken over p ramified.

I⋂

a,b>0 Ta,b = Z.For each p one must find pairs a, b, that ramify at p. This is not hard.



Poonen’s definition of Z in Q:

ψ(x) = ∀a∀b∃x1 . . . ∃x4∃y1 . . . ∃y4(a+x21 +x2

2 +x23 +x2

4 )(b+y21 +y2

2 +y23 +y2

4 )[(x1 − ax2

2 − bx23 + abx2

4 − 1)2 +∏2309

n=0 ((n − x − 2x1)2 − 4ay22 − 4by2

3 +4aby2

4 − 4)2] = 0.

RemarksThe top line has the sole function of allowing one to ignore negative valuesof a and b.The two universal quantifiers come from using local-global properties andit is hard to see how this approach would allow one to get down toexistential-only.It is however the first reduction in number of alternations of quantifiers in60 years.Notes after talk: Marker pointed out something I should have mentioned,that Rumely gave a proof of Julia Robinson’s result which is considerablyless “ad hoc” than hers, using local-global properties in a systematic way.



References

I K. Eisentrager, Integrality at a prime for global fields..., J. NumberTheory 114 (2005), 170-181.

I B. Poonen, Characterizing integers among rational numbers with auniversal existential formula, preprint (see also his article in the March2008 Notices of the AMS)

I J. Robinson, Definability of decision problems in arithmetic, J. S. L.14 (1949), 98-114. (See also Flath and Wagon’s Monthly article of1991, pp 812-823.

I R. Rumely, Undecidability and Definability for the Theory of GlobalFields, Trans AMS 262 (1980),pp. 195-217.

I A. Shlapentokh, Hilberts Tenth Problem, Cambridge UP 2007


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