Pseudorandomness of a Markoff Automorphism over F p Alois Cerbu - - PowerPoint PPT Presentation

▶

pseudorandomness of a markoff automorphism

Pseudorandomness of a Markoff Automorphism over F p Alois Cerbu - - PowerPoint PPT Presentation

Jul 03, 2023 43 likes •607 views

Pseudorandomness of a Markoff Automorphism over F p Alois Cerbu Elijah Gunther Luke Peilen Yale University 4 August, 2016 1 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p The Markoff

slide-1

SLIDE 1

Pseudorandomness of a Markoff Automorphism

ver Fp

Alois Cerbu Elijah Gunther Luke Peilen

Yale University

4 August, 2016

1 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-2

SLIDE 2

The Markoff Equation

2 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-3

SLIDE 3

The Markoff Equation

Classical Markoff Equation: x2 + y2 + z2 = 3xyz

2 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-4

SLIDE 4

The Markoff Equation

Classical Markoff Equation: x2 + y2 + z2 = 3xyz Variant: x2 + y2 + z2 = xyz

2 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-5

SLIDE 5

Solutions over a finite field

3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-6

SLIDE 6

Solutions over a finite field

Definition Define the variety over Fp V (Fp) = {(x, y, z) ∈ (Fp)3 | x2 + y2 + z2 = xyz}.

3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-7

SLIDE 7

Solutions over a finite field

Definition Define the variety over Fp V (Fp) = {(x, y, z) ∈ (Fp)3 | x2 + y2 + z2 = xyz}. Remark The size of this set is |V (Fp)| ≃ p2.

3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-8

SLIDE 8

Solutions over a finite field

Definition Define the variety over Fp V (Fp) = {(x, y, z) ∈ (Fp)3 | x2 + y2 + z2 = xyz}. Remark The size of this set is |V (Fp)| ≃ p2.

Note: We discard the “trivial” solution (0, 0, 0).

3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-9

SLIDE 9

Solutions over a finite field

Definition Define the variety over Fp V (Fp) = {(x, y, z) ∈ (Fp)3 | x2 + y2 + z2 = xyz}. Remark The size of this set is |V (Fp)| ≃ p2.

Note: We discard the “trivial” solution (0, 0, 0).

We will examine polynomial automorphisms of V (Fp).

3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-10

SLIDE 10

Vieta Involutions

4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-11

SLIDE 11

Vieta Involutions

(x − α)(x − β) = x2 − (α + β)x + αβ = 0

4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-12

SLIDE 12

Vieta Involutions

(x − α)(x − β) = x2 − (α + β)x + αβ = 0 x2 − (yz)x + (y2 + z2) = 0

4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-13

SLIDE 13

Vieta Involutions

(x − α)(x − β) = x2 − (α + β)x + αβ = 0 x2 − (yz)x + (y2 + z2) = 0 m1 :   x y z   − →   yz − x y z   ,

4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-14

SLIDE 14

Vieta Involutions

(x − α)(x − β) = x2 − (α + β)x + αβ = 0 x2 − (yz)x + (y2 + z2) = 0 m1 :   x y z   − →   yz − x y z   , m2 :   x y z   − →   x xz − y z   , m3 :   x y z   − →   x y xy − z  

4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-15

SLIDE 15

Even Sign Changes & S3

5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-16

SLIDE 16

Even Sign Changes & S3

x2 + y2 + z2 = xyz n1 :   x y z   − →   x −y −z  

5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-17

SLIDE 17

Even Sign Changes & S3

x2 + y2 + z2 = xyz n1 :   x y z   − →   x −y −z   n2 :   x y z   − →   −x y −z   n3 :   x y z   − →   −x −y z  

5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-18

SLIDE 18

Even Sign Changes & S3

x2 + y2 + z2 = xyz n1 :   x y z   − →   x −y −z   n2 :   x y z   − →   −x y −z   n3 :   x y z   − →   −x −y z   x1 := x, x2 := y, x3 := z; σ ∈ S3 acts by xi → xσ(i).

5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-19

SLIDE 19

Even Sign Changes & S3

x2 + y2 + z2 = xyz n1 :   x y z   − →   x −y −z   n2 :   x y z   − →   −x y −z   n3 :   x y z   − →   −x −y z   x1 := x, x2 := y, x3 := z; σ ∈ S3 acts by xi → xσ(i). Example (1 3 2) :   x1 x2 x3   − →   x2 x3 x1  

5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-20

SLIDE 20

The Automorphism Group, Γ

6 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-21

SLIDE 21

The Automorphism Group, Γ

Theorem (Horowitz, 1975) Vieta involutions, even sign changes, and permutations of the coordinates generate the full group Γ of polynomial automorphisms

f the variety.

6 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-22

SLIDE 22

The Automorphism Group, Γ

Theorem (Horowitz, 1975) Vieta involutions, even sign changes, and permutations of the coordinates generate the full group Γ of polynomial automorphisms

f the variety.

Conjecture (McCullough, Wanderley, 2013) Strong Approximation: The action of Γ on V (Fp) \ {(0, 0, 0)} is transitive for all primes.

6 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-23

SLIDE 23

The Automorphism Group, Γ

Theorem (Horowitz, 1975) Vieta involutions, even sign changes, and permutations of the coordinates generate the full group Γ of polynomial automorphisms

f the variety.

Conjecture (McCullough, Wanderley, 2013) Strong Approximation: The action of Γ on V (Fp) \ {(0, 0, 0)} is transitive for all primes. Theorem (Bourgain, Gamburd, Sarnak, 2016) The action of Γ on V (Fp) \ {(0, 0, 0)} is transitive for almost all primes (all but a small and slowly-growing exceptional set).

6 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-24

SLIDE 24

Reduced Variety

Remark N = n1, n2, n3 Γ

7 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-25

SLIDE 25

Reduced Variety

Remark N = n1, n2, n3 Γ Remark Γ acts on W(Fp), the set of N-blocks.

7 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-26

SLIDE 26

Reduced Variety

Remark N = n1, n2, n3 Γ Remark Γ acts on W(Fp), the set of N-blocks. Definition We denote by H(p) the permutation representation of this action.

7 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-27

SLIDE 27

Reduced Variety

Remark N = n1, n2, n3 Γ Remark Γ acts on W(Fp), the set of N-blocks. Definition We denote by H(p) the permutation representation of this action. The remainder of the talk concerns this series {H(p)} of finite permutation groups.

7 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-28

SLIDE 28

Permutation Group, H(p)

8 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-29

SLIDE 29

Permutation Group, H(p)

Let |W(Fp)| = n. Lemma (CGP 2016) H(p) ≤ An if and only p ≡ 3 (mod 16).

8 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-30

SLIDE 30

Permutation Group, H(p)

Let |W(Fp)| = n. Lemma (CGP 2016) H(p) ≤ An if and only p ≡ 3 (mod 16). Conjecture (CGP 2016) H(p) ∼ = Sn for p ≡ 3 (mod 16) H(p) ∼ = An for p ≡ 3 (mod 16) We have checked this for primes up to 31.

8 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-31

SLIDE 31

Pseudorandom Behavior

9 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-32

SLIDE 32

Pseudorandom Behavior

Question: Does a fixed automorphism behave pseudorandomly, modulo p?

9 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-33

SLIDE 33

Pseudorandom Behavior

Question: Does a fixed automorphism behave pseudorandomly, modulo p? γ ∈ Γ γ5 ∈ H(5) γ7 ∈ H(7) γ11 ∈ H(11) . . . γp ∈ H(p) . . . (Recall H(p) is the permutation group generated by the action of Γ on W(Fp))

9 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-34

SLIDE 34

Nonexamples

10 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-35

SLIDE 35

Nonexamples

Automorphisms m1, n1, σ ∈ S3. Don’t change all the coordinates or entries; have low order.

10 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-36

SLIDE 36

Nonexamples

Automorphisms m1, n1, σ ∈ S3. Don’t change all the coordinates or entries; have low order. This happens with probability zero.

10 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-37

SLIDE 37

Fixed Automorphism

11 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-38

SLIDE 38

Fixed Automorphism

We focus on one element of Γ acting on W(Fp): ϑ :   x y z   m1 − →   yz − x y z   (132) − →   y z yz − x  

11 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-39

SLIDE 39

Fixed Automorphism

We focus on one element of Γ acting on W(Fp): ϑ :   x y z   m1 − →   yz − x y z   (132) − →   y z yz − x   This automorphism preserves no obvious structure.

11 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-40

SLIDE 40

Counting Cycles

12 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-41

SLIDE 41

Counting Cycles

Fact For σ chosen uniformly at random from Sn or An, and k < n, the expected number of k-cycles in the decomposition for σ equals 1/k. Goal: Count the number of k-cycles in the decomposition of ϑp

ver all primes.

12 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-42

SLIDE 42

Counting Cycles

Fact For σ chosen uniformly at random from Sn or An, and k < n, the expected number of k-cycles in the decomposition for σ equals 1/k. Goal: Count the number of k-cycles in the decomposition of ϑp

ver all primes. If {ϑp} behaves as a family of random

permutations, we expect Ek := lim

x→∞

1 π(x)

p≤x

#{k-cycles in ϑp} = 1 k where π(x) is the prime counting function.

12 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-43

SLIDE 43

Counting Cycles: Proven, k ≤ 5

13 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-44

SLIDE 44

Counting Cycles: Proven, k ≤ 5

Theorem (CGP 2016) Let Ek be the expected number of k-cycles in the decomposition

f ϑp ∈ W(Fp). Then

k 1 2 3 4 5 Ek 1/2 1/2 1/5

13 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-45

SLIDE 45

Counting Cycles: Proven, k ≤ 5

Theorem (CGP 2016) Let Ek be the expected number of k-cycles in the decomposition

f ϑp ∈ W(Fp). Then

k 1 2 3 4 5 Ek 1/2 1/2 1/5 Proof. Idea of proofs: For k ≤ 4, quadratic reciprocity, polynomial manipulations, algebra. For k = 5, Galois Theory and Chebotarev Density Theorem.

13 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-46

SLIDE 46

Long Orbits

14 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-47

SLIDE 47

Long Orbits

Fact With positive probability, a random permutation σ ∈ Sn has an

rbit of length ≥ n/2.

14 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-48

SLIDE 48

Long Orbits

Fact With positive probability, a random permutation σ ∈ Sn has an

rbit of length ≥ n/2.

If ϑp indeed behaves pseudorandomly, we expect its longest orbits to grow linearly with |W(Fp)| ≃ p2.

14 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-49

SLIDE 49

Lower Bound

15 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-50

SLIDE 50

Lower Bound

We prove a lower bound for the size of the largest orbit of the action of ϑ on W(Fp). Theorem (CGP 2016) ∀ primes p, ∃ an orbit of ϑ on W(Fp) of length greater than N0 = log(p) log(ϕ) − κ, where κ is a positive constant and ϕ = 1+

√ 5 2

, the golden ratio.

15 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-51

SLIDE 51

Lower Bound

We prove a lower bound for the size of the largest orbit of the action of ϑ on W(Fp). Theorem (CGP 2016) ∀ primes p, ∃ an orbit of ϑ on W(Fp) of length greater than N0 = log(p) log(ϕ) − κ, where κ is a positive constant and ϕ = 1+

√ 5 2

, the golden ratio. Proof. Idea: Combinatorial argument, and the fact that a polynomial of degree n ≥ 1 has no more than n roots.

15 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-52

SLIDE 52

Acknowledgements

16 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-53

SLIDE 53

Acknowledgements

We’d like to thank the organizers of MathFest.

16 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-54

SLIDE 54

Acknowledgements

We’d like to thank the organizers of MathFest. Special thanks to our mentor, Michael Magee, and to the other

rganizers of the SUMRY REU, Jos´

e Gonzalez and Sam Payne.

16 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-55

SLIDE 55

Acknowledgements

We’d like to thank the organizers of MathFest. Special thanks to our mentor, Michael Magee, and to the other

rganizers of the SUMRY REU, Jos´

e Gonzalez and Sam Payne. Thank you!

16 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp

slide-56

SLIDE 56

Questions

17 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over Fp