Sums of Two
Squares inFunction
FieldsBary-Soroker

Sums of Two Squares in Function Fields

Sums of Two
SquaresClassical Theory
An Open Problem

FunctionFields
Classical Theory

Lior Bary-Soroker
Tel Aviv University

Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Franco-German Summer School
Galois Theory and Number Theory
Konstanz, July 2015

1/18

Outline
Sums of Two
Squares inFunction
FieldsBary-Soroker

1

Sums of Two Squares
Classical Theory
An Open Problem

2

Function Fields
Classical Theory
Short Intervals

3

Methods of Proof
A Galois Description of bq (f ) = 1
Equidistribution

Sums of Two
SquaresClassical Theory
An Open Problem

FunctionFields
Classical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

2/18

Outline
Sums of Two
Squares inFunction
FieldsBary-Soroker

1

Sums of Two Squares
Classical Theory
An Open Problem

2

Function Fields
Classical Theory
Short Intervals

3

Methods of Proof
A Galois Description of bq (f ) = 1
Equidistribution

Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

3/18

Basic Algebraic Theory
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsCharacteristic Function(1, n =  + b(n) =0, otherwise.
Examples: b(1) = b(2) = b(4) = b(5) = 1,
while b(3) = b(7) = 0

Classical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1

Description as Norms of Gaussian integersb(n) = 1 ⇐⇒ n = NormZ[i] (a + bi)

Equidistribution

Multiplicative Description – Fermat’s TheoremYYb(n) = 1 ⇐⇒ N = 2αpeq 2ep≡1(4)q≡3(4)
4/18

Asymptotic Density
Landau’s Theorem
Sums of TwoSquares inFunction
FieldsBary-Soroker

We denote byhϕ(n)in∈I =Sums of TwoSquares
Classical Theory
An Open Problem

Function
FieldsClassical Theory

1 X
ϕ(n)
|I|n∈I

the average of the function ϕ on the set I
Theorem (Landau)

Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1

K
hb(n)in≤x = p+O
log x



1(log x)3/2

,x →∞

Equidistribution

Q−2 −1/2 ≈ 0.764 is theHere K = √1p≡3(4) (1 − p )2Landau-Ramanujan constant
5/18

Density in Short Intervals
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory

Naive Expectation
Khb(n)i|n−x|≤φ(x) ∼ hb(n)in≤x ∼ plog xppfor log x < φ(x) < x with φ(x)/ log x → ∞.

Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

6/18

Density in Short Intervals
Sums of Two
Squares inFunctionFields
Bary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory

Naive Expectation
Khb(n)i|n−x|≤φ(x) ∼ hb(n)in≤x ∼ plog xpp
for log x < φ(x) < x with φ(x)/ log x → ∞.

Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Open Problem
How small can φ(x) be so that the naive expectation will
hold?

6/18

History of the Problem
Sums of Two
Squares inFunction
Fields

How small can φ(x) be so that hb(n)i|n−x|≤φ(x) ∼ √ Klog x?

Bary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

7/18

History of the Problem
Sums of Two
Squares inFunction
Fields

How small can φ(x) be so that hb(n)i|n−x|≤φ(x) ∼ √ K

log x?

Bary-Soroker
Sums of Two
SquaresClassical Theory

Not Too Small
Balog-Wooley: Fails for any φ(x) = (log x)A

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

7/18

History of the Problem
Sums of Two
Squares inFunction
Fields

How small can φ(x) be so that hb(n)i|n−x|≤φ(x) ∼ √ Klog x?

Bary-Soroker
Sums of Two
SquaresClassical Theory

Not Too Small
Balog-Wooley: Fails for any φ(x) = (log x)A

An Open Problem

Function
FieldsBest Result

Classical Theory
Short Intervals

Huxely, Heath-Brown (methods from primes): φ(x) ≥ x 7/12

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

7/18

History of the Problem
Sums of Two
Squares inFunctionFields

How small can φ(x) be so that hb(n)i|n−x|≤φ(x) ∼ √ Klog x?

Bary-Soroker
Sums of TwoSquares
Classical Theory

Not Too Small
Balog-Wooley: Fails for any φ(x) = (log x)A

An Open Problem

Function
FieldsBest Result

Classical Theory
Short Intervals

Huxely, Heath-Brown (methods from primes): φ(x) ≥ x 7/12

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Sieve Methods
Iwaniec (1976)
Hooley (1974, 1994), Friedlander (1982),
Plaskin (1987), Harman (1991)
Balog-Wooley (2000)
7/18

Folklore Conjecture
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory

Conjecture
Let  > 0. Then

An Open Problem

Function
FieldsClassical Theory

Khb(n)i|n−x|≤x  ∼ hb(n)in≤x ∼ p,
log x

x →∞

Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

8/18

Folklore Conjecture
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory

Conjecture
Let  > 0. Then

An Open Problem

Function
FieldsClassical Theory

Khb(n)i|n−x|≤x  ∼ hb(n)in≤x ∼ p,log xx →∞Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Since this conjecture is completely open we will study it infunction fields

8/18

Outline
Sums of Two
Squares inFunction
FieldsBary-Soroker

1

Sums of Two Squares
Classical Theory
An Open Problem

2

Function Fields
Classical Theory
Short Intervals

3

Methods of Proof
A Galois Description of bq (f ) = 1
Equidistribution

Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1Equidistribution

9/18

What is the Analogue of Sums of Two Squares?
Sums of Two
Squares inFunction
FieldsInstead of Z we work with Fq [T ] with q odd

Bary-Soroker

f (T ) = A(T )2 + B 2 (T )Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

10/18

What is the Analogue of Sums of Two Squares?
Sums of Two
Squares inFunction
FieldsInstead of Z we work with Fq [T ] with q odd

Bary-Soroker

f (T ) = A(T )2 + B 2 (T ), e.g. = (A + 2B)(A − 2B) mod 5
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

10/18

What is the Analogue of Sums of Two Squares?
Sums of Two
Squares inFunction
FieldsInstead of Z we work with Fq [T ] with q odd

Bary-Soroker

f (T ) = A(T )2 + B 2 (T ), e.g. = (A + 2B)(A − 2B) mod 5
Sums of Two
SquaresClassical Theory
An Open Problem

FunctionFields√f (T ) = A(T )2 + TB 2 (T ) – a norm from Fq [ −T ]/Fq [T ]bq (f ) = 1 in this case, otherwise bq (f ) = 0

Classical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

10/18

What is the Analogue of Sums of Two Squares?
Sums of Two
Squares inFunction
FieldsInstead of Z we work with Fq [T ] with q odd

Bary-Soroker

f (T ) = A(T )2 + B 2 (T ), e.g. = (A + 2B)(A − 2B) mod 5
Sums of TwoSquares
Classical Theory
An Open Problem

FunctionFields√f (T ) = A(T )2 + TB 2 (T ) – a norm from Fq [ −T ]/Fq [T ]bq (f ) = 1 in this case, otherwise bq (f ) = 0

Classical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Multiplicative description – "Fermat’s Theorem"
YYbq (f ) = 1 ⇐⇒ f = T α
P eP
Q 2eQ
where T 6= P (resp. Q) runs over all prime polynomials withP(−T 2 ) (resp. Q(−T 2 )) reducible (resp. irreducible)

10/18Function Field Landau’s Theorem
Two Limits Phenomenon
Sums of Two
Squares inFunction
FieldsBary-Soroker

Denote Mn,q ⊆ Fq [T ], monic polynomials of degree nMn,q ! {n : 1 ≤ n ≤ x}

and #Mn,q = q n ! x

Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

11/18

Function Field Landau’s Theorem
Two Limits Phenomenon
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory

Denote Mn,q ⊆ Fq [T ], monic polynomials of degree nMn,q ! {n : 1 ≤ n ≤ x}

and #Mn,q = q n ! xTheorem (BS-Smilanski-Wolf)

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Kq
hbq (f )if ∈Mn,q = √ + Oq (n−3/2 ),n2nhbq (f )if ∈Mn,q =n4n+ On (q −1 ),

n→∞
q→∞

11/18

Function Field Landau’s Theorem
Two Limits Phenomenon
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of TwoSquares
Classical Theory

Denote Mn,q ⊆ Fq [T ], monic polynomials of degree nMn,q ! {n : 1 ≤ n ≤ x}

and #Mn,q = q n ! x

Theorem (BS-Smilanski-Wolf)

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Kq
hbq (f )if ∈Mn,q = √ + Oq (n−3/2 ),n
2nhbq (f )if ∈Mn,q =n4n+ On (q −1 ),

n→∞
q→∞

Comparison of main terms in the different limits
Kq 4n
Kq 4nlim lim √ 2n = lim lim √ 2n
n→∞ q→∞
q→∞ n→∞n nn n
11/18

Main Result
Large Finite Field Limit
Sums of Two
Squares inFunction
FieldsConjecture: hb(n)i|n−x|≤x  ∼ hb(n)in≤x ∼ √ Klog x

Bary-Soroker
Sums of Two
SquaresClassical Theory

Norm of a polynomial: |f | = #(Fq [T ]/f ) = q deg f
Analogue of |n − x| ≤ x  : |f − f0 | ≤ |f0 | , f0 ∈ Mn,q

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProofA Galois Descriptionof bq (f ) = 1Equidistribution

12/18

Main Result
Large Finite Field Limit
Sums of Two
Squares inFunction
FieldsConjecture: hb(n)i|n−x|≤x  ∼ hb(n)in≤x ∼ √ Klog x

Bary-Soroker
Sums of Two
SquaresClassical Theory

Norm of a polynomial: |f | = #(Fq [T ]/f ) = q deg f
Analogue of |n − x| ≤ x  : |f − f0 | ≤ |f0 | , f0 ∈ Mn,q

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Theorem (Bank-BS-Fehm)
Fix n and n2 ≤  < 1. Then2nn4n

hbq (f )i|f −f0 |≤|f0 | =+ On (q −1/2 ),q→∞uniformly on f0 ∈ Mn,q .

12/18

Main Result
Large Finite Field Limit
Sums of Two
Squares inFunction
FieldsConjecture: hb(n)i|n−x|≤x  ∼ hb(n)in≤x ∼ √ K

log x

Bary-Soroker
Sums of Two
SquaresClassical Theory

Norm of a polynomial: |f | = #(Fq [T ]/f ) = q deg f
Analogue of |n − x| ≤ x  : |f − f0 | ≤ |f0 | , f0 ∈ Mn,qAn Open Problem

Function
FieldsClassical TheoryShort Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Theorem (Bank-BS-Fehm)
Fix n and n2 ≤  < 1. Then2nn4n

hbq (f )i|f −f0 |≤|f0 | =+ On (q −1/2 ),q→∞uniformly on f0 ∈ Mn,q .
Completely settles the function field conjecture for large q
12/18

Outline
Sums of Two
Squares inFunction
FieldsBary-Soroker

1

Sums of Two Squares
Classical Theory
An Open Problem

2

Function Fields
Classical Theory
Short Intervals

3

Methods of Proof
A Galois Description of bq (f ) = 1Equidistribution

Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

13/18

The Hyperoctahedral Group as Galois GroupOver a General Field
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Observation
QLet f (T ) = ni=1 (T +Qyi ) and√√g(T ) = f (−T 2 ) = ± ni=1 (T − yi )(T + yi ) be squarefreepolynomials over a field K of characteristic 6= 2. Then

FunctionFields

Gal(g(T )/K ) ,→ Hn

Classical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Hn = Aut

14/18

The Hyperoctahedral Group as Galois GroupOver a Finite Field
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theoryg(T ) = f (−T 2 ) =Q

f (−yi )=0 (T −√Gal(g(T )/Fq ) = hφq i ,yi )(T +√yi )φq (x) = x q

So g gives an element φq,g ∈ Hn .

An Open Problem

Function
FieldsClassical Theory
Short IntervalsMethods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Proposition
There exists Xn ⊆ Hn such that bq (f ) = 1 if and only if
φq ∈ X n
15/18

The Hyperoctahedral Group as Galois GroupOver a Finite Field
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theoryg(T ) = f (−T 2 ) =Q

f (−yi )=0 (T −√Gal(g(T )/Fq ) = hφq i ,yi )(T +√yi )φq (x) = x q

So g gives an element φq,g ∈ Hn .

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Proposition
There exists Xn ⊆ Hn such that bq (f ) = 1 if and only ifφq ∈ X n15/18

The Hyperoctahedral Group as Galois GroupOver a Finite Field
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theoryg(T ) = f (−T 2 ) =Q

f (−yi )=0 (T −√Gal(g(T )/Fq ) = hφq i ,yi )(T +√yi )φq (x) = x q

So g gives an element φq,g ∈ Hn .

An Open ProblemFunction
FieldsClassical Theory
Short Intervals

Methods ofProofA Galois Descriptionof bq (f ) = 1
Equidistribution

Proposition
There exists Xn ⊆ Hn such that bq (f ) = 1 if and only ifφq ∈ X n
15/18

The Hyperoctahedral Group as Galois GroupOver a Finite Field
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theoryg(T ) = f (−T 2 ) =Q

f (−yi )=0 (T −√Gal(g(T )/Fq ) = hφq i ,yi )(T +√yi )φq (x) = x q

So g gives an element φq,g ∈ Hn .

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

Proposition
There exists Xn ⊆ Hn such that bq (f ) = 1 if and only if
φq ∈ X n
15/18

Explicit Chebotarev’s Theorem
Sums of Two
Squares inFunction
FieldsFor f0 ∈ Mn,q put

Bary-Soroker

FA (T ) = f0 (T ) +Sums of TwoSquares

X

Ai T ii≤nClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1Equidistribution

with A = (A0 , . . . , Am ) a tuple of variables (m = bnc). Let
GA (T ) = FA (−T 2 )and G = Gal(GA ) ⊆ Hn

Theorembnc+1

Prob(a ∈ Fq: φq,Ga ∈ Xn ) ∼ ProbG (σ ∈ Xn )Note that LHS = hb(f )i|f −f0 |≤|f0 |
16/18

Conclusion of the Proof
Sums of Two
Squares inFunction
Fields

G = Gal(GA (T )) = Gal(FA (−T 2 )) ⊆ Hn
Key Proposition

Bary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1

G = Hn(Builds on Bank-BS-Rosenzweig: Gal(FA (T )) = Sn )Corollary #Xn
1 2n= nhb(f )i|f −f0 |≤|f0 | ∼#Hn
4n

Equidistribution

17/18

Conclusion of the Proof
Sums of Two
Squares inFunction
Fields

G = Gal(GA (T )) = Gal(FA (−T 2 )) ⊆ Hn
Key Proposition

Bary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1

G = Hn(Builds on Bank-BS-Rosenzweig: Gal(FA (T )) = Sn )Corollary #Xn
1 2n= nhb(f )i|f −f0 |≤|f0 | ∼#Hn
4n

Equidistribution

Remarks
This completes the proof

17/18

Conclusion of the Proof
Sums of Two
Squares inFunction
Fields

G = Gal(GA (T )) = Gal(FA (−T 2 )) ⊆ Hn
Key Proposition

Bary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1

G = Hn(Builds on Bank-BS-Rosenzweig: Gal(FA (T )) = Sn )Corollary #Xn
1 2n= nhb(f )i|f −f0 |≤|f0 | ∼#Hn4n

Equidistribution

Remarks
This completes the proof
The last inequality comes from Ewens’ sampling
formula17/18

Further Problems to Think About
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

18/18

Further Problems to Think About
Sums of Two
Squares inFunction
FieldsLimit n → ∞Bary-Soroker
Sums of Two
SquaresClassical Theory
An Open Problem

FunctionFields
Classical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

18/18

Further Problems to Think About
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of TwoSquares
Classical Theory

Limit n → ∞Further study of the error term
 
1 2n
E = hbq (f )i|f −f0 |≤|f0 | − n4n

An Open Problem

Function
Fields(Sharper upper bounds on |E|, average, variance,...)Classical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1
Equidistribution

18/18

Further Problems to Think About
Sums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory

Limit n → ∞Further study of the error term 1 2nE = hbq (f )i|f −f0 |≤|f0 | − n4n

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1(Sharper upper bounds on |E|, average, variance,...)
Obtaining main terms in different problems, e.g.
hbq (f )bq (f + h)if ∈Mn,q

Equidistribution

18/18

Further Problems to Think AboutSums of Two
Squares inFunction
FieldsBary-Soroker
Sums of Two
SquaresClassical Theory

Limit n → ∞Further study of the error term 1 2nE = hbq (f )i|f −f0 |≤|f0 | − n4n

An Open Problem

Function
FieldsClassical Theory
Short Intervals

Methods ofProof
A Galois Descriptionof bq (f ) = 1(Sharper upper bounds on |E|, average, variance,...)
Obtaining main terms in different problems, e.g.
hbq (f )bq (f + h)if ∈Mn,q

Equidistribution

Landau’s theorem in q n → ∞Some progress was already obtained by Ofir
Gorodetsky
18/18