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Properties of Laurent coefficients of multivariate rational functions Workshop on Computer Algebra in Combinatorics Erwin Schroedinger Institute Armin Straub November 14, 2017 University of South Alabama includes joint work with and Frits

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SLIDE 1

Properties of Laurent coefficients

  • f multivariate rational functions

Workshop on Computer Algebra in Combinatorics Erwin Schroedinger Institute Armin Straub November 14, 2017 University of South Alabama includes joint work with and

Frits Beukers Marc Houben Wadim Zudilin

(Utrecht University) (Utrecht University) (University of Newcastle/ Radboud Universiteit)

Properties of Laurent coefficients of multivariate rational functions Armin Straub 1 / 34

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SLIDE 2

Goal of this talk

We introduce and advertise two questions about rational functions like 1 1 − (x1 + x2 + x3) + 4x1x2x3 =

  • n∈Z3

A(n)xn.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

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SLIDE 3

Goal of this talk

We introduce and advertise two questions about rational functions like 1 1 − (x1 + x2 + x3) + 4x1x2x3 =

  • n∈Z3

A(n)xn. When has a rational function the Gauss property? That is, when do the following congruences hold? A(npr) ≡ A(npr−1) (mod pr)

Q

Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

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SLIDE 4

Goal of this talk

We introduce and advertise two questions about rational functions like 1 1 − (x1 + x2 + x3) + 4x1x2x3 =

  • n∈Z3

A(n)xn. When has a rational function the Gauss property? That is, when do the following congruences hold? A(npr) ≡ A(npr−1) (mod pr)

Q

When is a rational function positive? That is, when is A(n) > 0 for all n?

Q

Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

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SLIDE 5

Goal of this talk

We introduce and advertise two questions about rational functions like 1 1 − (x1 + x2 + x3) + 4x1x2x3 =

  • n∈Z3

A(n)xn. When has a rational function the Gauss property? That is, when do the following congruences hold? A(npr) ≡ A(npr−1) (mod pr)

Q

When is a rational function positive? That is, when is A(n) > 0 for all n?

Q

In both cases, we will wonder about an explicit characterization.

These are not conjectures because our evidence is limited. Computer algebra!

Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

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SLIDE 6

Goal of this talk

We introduce and advertise two questions about rational functions like 1 1 − (x1 + x2 + x3) + 4x1x2x3 =

  • n∈Z3

A(n)xn. Here, the diagonal coefficients are the Franel numbers A(n, n, n) =

n

  • k=0

n k 3 .

EG

  • As seen in previous talks, simple multivariate generating functions can

be enormously useful, for instance, in computing asymptotics.

  • Time permitting, more on Ap´

ery-like numbers later. . .

Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

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SLIDE 7

I

Gauss congruences

Properties of Laurent coefficients of multivariate rational functions Armin Straub 3 / 34

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SLIDE 8

The classical Gauss congruence

ap ≡ a (mod p)

if p is prime.

THM

Fermat

Properties of Laurent coefficients of multivariate rational functions Armin Straub 4 / 34

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SLIDE 9

The classical Gauss congruence

ap ≡ a (mod p)

if p is prime.

THM

Fermat

aφ(m) ≡ 1 (mod m)

if a is coprime to m.

THM

Euler

Properties of Laurent coefficients of multivariate rational functions Armin Straub 4 / 34

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SLIDE 10

The classical Gauss congruence

ap ≡ a (mod p)

if p is prime.

THM

Fermat

aφ(m) ≡ 1 (mod m)

if a is coprime to m.

THM

Euler

  • d|m

µ( m

d )ad ≡ 0

(mod m)

THM

Gauss

M¨

  • bius function: µ(n) = (−1)# of p|n if n is square-free, µ(n) = 0 else

Properties of Laurent coefficients of multivariate rational functions Armin Straub 4 / 34

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SLIDE 11

The classical Gauss congruence

ap ≡ a (mod p)

if p is prime.

THM

Fermat

aφ(m) ≡ 1 (mod m)

if a is coprime to m.

THM

Euler

  • d|m

µ( m

d )ad ≡ 0

(mod m)

THM

Gauss

M¨

  • bius function: µ(n) = (−1)# of p|n if n is square-free, µ(n) = 0 else

If m = pr then only d = pr, d = pr−1 contribute, and we get apr ≡ apr−1 (mod pr).

EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 4 / 34

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Gauss congruences

a(n) satisfies the Gauss congruences if, for all primes p, a(mpr) ≡ a(mpr−1) (mod pr).

DEF

Equivalently,

  • d|m

µ( m

d )a(d) ≡ 0

(mod m).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

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SLIDE 13

Gauss congruences

a(n) satisfies the Gauss congruences if, for all primes p, a(mpr) ≡ a(mpr−1) (mod pr).

DEF

Equivalently,

  • d|m

µ( m

d )a(d) ≡ 0

(mod m).

  • a(n) = an

EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

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SLIDE 14

Gauss congruences

a(n) satisfies the Gauss congruences if, for all primes p, a(mpr) ≡ a(mpr−1) (mod pr).

DEF

Equivalently,

  • d|m

µ( m

d )a(d) ≡ 0

(mod m).

  • a(n) = an
  • a(n) = Ln

Lucas numbers:

Ln+1 = Ln + Ln−1 L0 = 2, L1 = 1 EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

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SLIDE 15

Gauss congruences

a(n) satisfies the Gauss congruences if, for all primes p, a(mpr) ≡ a(mpr−1) (mod pr).

DEF

Equivalently,

  • d|m

µ( m

d )a(d) ≡ 0

(mod m).

  • a(n) = an
  • a(n) = Ln

Lucas numbers:

Ln+1 = Ln + Ln−1 L0 = 2, L1 = 1

  • a(n) = Dn

Delannoy numbers: Dn =

n

  • k=0

n k n + k k

  • EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

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SLIDE 16

Gauss congruences

a(n) satisfies the Gauss congruences if, for all primes p, a(mpr) ≡ a(mpr−1) (mod pr).

DEF

Equivalently,

  • d|m

µ( m

d )a(d) ≡ 0

(mod m).

  • a(n) = an
  • a(n) = Ln

Lucas numbers:

Ln+1 = Ln + Ln−1 L0 = 2, L1 = 1

  • a(n) = Dn

Delannoy numbers: Dn =

n

  • k=0

n k n + k k

  • EG
  • Later, we allow a(n) ∈ Q. If the Gauss congruences hold for all but finitely

many p, we say that the sequence (or its GF) has the Gauss property.

  • Similarly, for multivariate sequences a(n), we require

a(mpr) ≡ a(mpr−1) (mod pr).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

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More sequences satisfying Gauss congruences

a(mpr) ≡ a(mpr−1) (mod pr) (G)

  • realizable sequences a(n), i.e., for some map T : X → X,

a(n) = #{x ∈ X : T nx = x} “points of period n”

Everest–van der Poorten–Puri–Ward ’02, Arias de Reyna ’05

In fact, up to a positivity condition, (G) characterizes realizability.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 6 / 34

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More sequences satisfying Gauss congruences

a(mpr) ≡ a(mpr−1) (mod pr) (G)

  • realizable sequences a(n), i.e., for some map T : X → X,

a(n) = #{x ∈ X : T nx = x} “points of period n”

Everest–van der Poorten–Puri–Ward ’02, Arias de Reyna ’05

In fact, up to a positivity condition, (G) characterizes realizability.

  • a(n) = trace(Mn)

J¨ anichen ’21, Schur ’37; also: Arnold, Zarelua

where M is an integer matrix

Properties of Laurent coefficients of multivariate rational functions Armin Straub 6 / 34

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SLIDE 19

More sequences satisfying Gauss congruences

a(mpr) ≡ a(mpr−1) (mod pr) (G)

  • realizable sequences a(n), i.e., for some map T : X → X,

a(n) = #{x ∈ X : T nx = x} “points of period n”

Everest–van der Poorten–Puri–Ward ’02, Arias de Reyna ’05

In fact, up to a positivity condition, (G) characterizes realizability.

  • a(n) = trace(Mn)

J¨ anichen ’21, Schur ’37; also: Arnold, Zarelua

where M is an integer matrix

  • (G) is equivalent to exp

∞

  • n=1

a(n) n T n

  • ∈ Z[[T]].

This is a natural condition in formal group theory.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 6 / 34

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SLIDE 20

Minton’s theorem

f ∈ Q(x) has the Gauss property if and only if f is a Q-linear combination of functions xu′(x)/u(x), with u ∈ Z[x].

THM

Minton, 2014

Properties of Laurent coefficients of multivariate rational functions Armin Straub 7 / 34

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SLIDE 21

Minton’s theorem

f ∈ Q(x) has the Gauss property if and only if f is a Q-linear combination of functions xu′(x)/u(x), with u ∈ Z[x].

THM

Minton, 2014

  • If u(x) = s

i=1(1 − αix) then

xu′(x) u(x) = −

s

  • i=1

αix 1 − αix = s −

s

  • i=1

1 1 − αix.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 7 / 34

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SLIDE 22

Minton’s theorem

f ∈ Q(x) has the Gauss property if and only if f is a Q-linear combination of functions xu′(x)/u(x), with u ∈ Z[x].

THM

Minton, 2014

  • If u(x) = s

i=1(1 − αix) then

xu′(x) u(x) = −

s

  • i=1

αix 1 − αix = s −

s

  • i=1

1 1 − αix.

  • Assuming the αi are distinct,

s

  • i=1

1 1 − αix =

  • n0

s

  • i=1

αn

i

  • xn =
  • n0

trace(Mn)xn,

where M is the companion matrix of s

i=1(x − αi) = xsu(1/x).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 7 / 34

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SLIDE 23

Minton’s theorem

f ∈ Q(x) has the Gauss property if and only if f is a Q-linear combination of functions xu′(x)/u(x), with u ∈ Z[x].

THM

Minton, 2014

  • If u(x) = s

i=1(1 − αix) then

xu′(x) u(x) = −

s

  • i=1

αix 1 − αix = s −

s

  • i=1

1 1 − αix.

  • Assuming the αi are distinct,

s

  • i=1

1 1 − αix =

  • n0

s

  • i=1

αn

i

  • xn =
  • n0

trace(Mn)xn,

where M is the companion matrix of s

i=1(x − αi) = xsu(1/x).

  • Minton: No new C-finite sequences with the Gauss property!
  • Can we generalize from C-finite towards D-finite?

Properties of Laurent coefficients of multivariate rational functions Armin Straub 7 / 34

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The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017 Interesting detail: true for any of the different Laurent expansions of multivariate rational functions

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 25

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017 Interesting detail: true for any of the different Laurent expansions of multivariate rational functions

Consider Q = 1 − x − y − z + 4xyz:

f1 = Q = ⇒ (D) = −x + 4xyz Q

EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 26

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017 Interesting detail: true for any of the different Laurent expansions of multivariate rational functions

Consider Q = 1 − x − y − z + 4xyz:

f1 = Q = ⇒ (D) = −x + 4xyz Q f1 = Q, f2 = 1 − 4yz = ⇒ (D) = 4xyz Q

EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 27

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017 Interesting detail: true for any of the different Laurent expansions of multivariate rational functions

Consider Q = 1 − x − y − z + 4xyz:

f1 = Q = ⇒ (D) = −x + 4xyz Q f1 = Q, f2 = 1 − 4yz = ⇒ (D) = 4xyz Q

In particular, 1 1 − x − y − z + 4xyz has the Gauss property.

EG

There is nothing special about 4.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 28

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017

Let P, Q ∈ Z[x] with Q is linear in each variable. Then P/Q has the Gauss property if and only if N(P) ⊆ N(Q).

THM

BHS

  • Here, N(Q) is the Newton polytope of Q.
  • In this case, N(Q) = supp(Q) ⊆ {0, 1}n.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 29

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017

Let P, Q ∈ Z[x] with Q is linear in each variable. Then P/Q has the Gauss property if and only if N(P) ⊆ N(Q).

THM

BHS

  • Here, N(Q) is the Newton polytope of Q.
  • In this case, N(Q) = supp(Q) ⊆ {0, 1}n.

Let P, Q ∈ Z[x±1]. If P/Q has the Gauss property, then N(P) ⊆ N(Q).

PROP

BHS

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 30

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017

Suppose f ∈ Q(x) has the Gauss property. Can it be written as a Q-linear combination of functions of the form (D)?

Q

BHS

  • Yes, for n = 1, by Minton’s theorem.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 31

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017

Suppose f ∈ Q(x) has the Gauss property. Can it be written as a Q-linear combination of functions of the form (D)?

Q

BHS

  • Yes, for n = 1, by Minton’s theorem.
  • Yes, for f = P/Q with Q linear in all, or all but one, variables.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 32

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017

Suppose f ∈ Q(x) has the Gauss property. Can it be written as a Q-linear combination of functions of the form (D)?

Q

BHS

  • Yes, for n = 1, by Minton’s theorem.
  • Yes, for f = P/Q with Q linear in all, or all but one, variables.
  • Yes, for f = P/Q with Q in two variables and total degree 2.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 33

The multivariate case

Let f1, . . . , fm ∈ Q(x) = Q(x1, . . . , xn) be nonzero. Then x1 · · · xm f1 · · · fm det ∂fj ∂xi

  • i,j=1,...,m

(D) has the Gauss property.

THM

Beukers, Houben, S 2017

Suppose f ∈ Q(x) has the Gauss property. Can it be written as a Q-linear combination of functions of the form (D)?

Q

BHS

  • Yes, for n = 1, by Minton’s theorem.
  • Yes, for f = P/Q with Q linear in all, or all but one, variables.
  • Yes, for f = P/Q with Q in two variables and total degree 2.

Can

x(x + y + y2 + 2xy2) 1 + 3x + 3y + 2x2 + 2y2 + xy − 2x2y2 be written in that form? EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 8 / 34

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SLIDE 34

Application: Delannoy numbers

Let P, Q ∈ Z[x] with Q is linear in each variable. Then P/Q has the Gauss property if and only if N(P) ⊆ N(Q).

THM

BHS

Properties of Laurent coefficients of multivariate rational functions Armin Straub 9 / 34

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SLIDE 35

Application: Delannoy numbers

Let P, Q ∈ Z[x] with Q is linear in each variable. Then P/Q has the Gauss property if and only if N(P) ⊆ N(Q).

THM

BHS

The Delannoy numbers Dn1,n2 are characterized by 1 1 − x − y − xy =

∞

  • n1,n2=0

Dn1,n2xn1yn2.

EG

Beukers, Houben, S 2017

Properties of Laurent coefficients of multivariate rational functions Armin Straub 9 / 34

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SLIDE 36

Application: Delannoy numbers

Let P, Q ∈ Z[x] with Q is linear in each variable. Then P/Q has the Gauss property if and only if N(P) ⊆ N(Q).

THM

BHS

The Delannoy numbers Dn1,n2 are characterized by 1 1 − x − y − xy =

∞

  • n1,n2=0

Dn1,n2xn1yn2. By the theorem, the following have the Gauss property: N 1 − x − y − xy with N ∈ {1, x, y, xy}

EG

Beukers, Houben, S 2017

Properties of Laurent coefficients of multivariate rational functions Armin Straub 9 / 34

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SLIDE 37

Application: Delannoy numbers

Let P, Q ∈ Z[x] with Q is linear in each variable. Then P/Q has the Gauss property if and only if N(P) ⊆ N(Q).

THM

BHS

The Delannoy numbers Dn1,n2 are characterized by 1 1 − x − y − xy =

∞

  • n1,n2=0

Dn1,n2xn1yn2. By the theorem, the following have the Gauss property: N 1 − x − y − xy with N ∈ {1, x, y, xy} In other words, for δ ∈ {0, 1}2, Dmpr−δ ≡ Dmpr−1−δ (mod pr).

EG

Beukers, Houben, S 2017

Properties of Laurent coefficients of multivariate rational functions Armin Straub 9 / 34

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SLIDE 38

II

Positivity

Properties of Laurent coefficients of multivariate rational functions Armin Straub 10 / 34

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SLIDE 39

Positivity of rational functions

  • A rational function

F(x1, . . . , xd) =

  • n1,...,nd0

an1,...,ndxn1

1 · · · xnd d

is positive if an1,...,nd > 0 for all indices.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 11 / 34

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SLIDE 40

Positivity of rational functions

  • A rational function

F(x1, . . . , xd) =

  • n1,...,nd0

an1,...,ndxn1

1 · · · xnd d

is positive if an1,...,nd > 0 for all indices.

1 1 − x and 1 (1 − x)(1 − y) are positive. EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 11 / 34

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SLIDE 41

Positivity of rational functions

  • A rational function

F(x1, . . . , xd) =

  • n1,...,nd0

an1,...,ndxn1

1 · · · xnd d

is positive if an1,...,nd > 0 for all indices.

1 1 − x and 1 (1 − x)(1 − y) are positive. EG

1 (1 − x)(1 − y) + (1 − y)(1 − z) + (1 − z)(1 − x) is positive. EG

Szeg˝

  • 1933
  • Szeg˝
  • ’s proof builds on an integral of a product of Bessel functions.

“the used tools, however, are disproportionate to the simplicity of the statement”

  • Elementary proof by Kaluza (’33)
  • Askey–Gasper (’72) use integral of product of Legendre functions.
  • Ismail–Tamhankar (’79) systematize Kaluza’s approach by using

MacMahon’s Master Theorem.

  • S (’08): simple proof using a positivity-preserving operator

Properties of Laurent coefficients of multivariate rational functions Armin Straub 11 / 34

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SLIDE 42

Historical motivation

1 (1 − x)(1 − y) + (1 − y)(1 − z) + (1 − z)(1 − x) =

  • k,m,n

A(k, m, n)xkymzn

  • Friedrichs and Lewy conjectured positivity of A(k, m, n).
  • Wanted to show convergence of finite difference approximations to

∂ ∂x ∂ ∂y + ∂ ∂x ∂ ∂z + ∂ ∂y ∂ ∂z

  • u(x, y, z) = 0,

which transforms to the 2D wave equation.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 12 / 34

slide-43
SLIDE 43

Historical motivation

1 (1 − x)(1 − y) + (1 − y)(1 − z) + (1 − z)(1 − x) =

  • k,m,n

A(k, m, n)xkymzn

  • Friedrichs and Lewy conjectured positivity of A(k, m, n).
  • Wanted to show convergence of finite difference approximations to

∂ ∂x ∂ ∂y + ∂ ∂x ∂ ∂z + ∂ ∂y ∂ ∂z

  • u(x, y, z) = 0,

which transforms to the 2D wave equation.

  • With ∂/∂x replaced by ∆k,

∆a(k) = a(k) − a(k − 1)

(∆k∆m + ∆k∆n + ∆m∆n)A(k, m, n) = 0.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 12 / 34

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SLIDE 44

Generalizations

  • Szeg˝
  • also showed positivity of

(and extension to any # of variables)

1 4

i=1

  • j=i(1 − xj)

= 1 (1 − x2)(1 − x3)(1 − x4) + · · · + (1 − x1)(1 − x2)(1 − x3)

Properties of Laurent coefficients of multivariate rational functions Armin Straub 13 / 34

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SLIDE 45

Generalizations

  • Szeg˝
  • also showed positivity of

(and extension to any # of variables)

1 4

i=1

  • j=i(1 − xj)

= 1 (1 − x2)(1 − x3)(1 − x4) + · · · + (1 − x1)(1 − x2)(1 − x3)

  • The Lewy–Askey problem asks for positivity of

1

  • 1i<j4

(1 − xi)(1 − xj) = 1 (1 − x1)(1 − x2) + · · · + (1 − x3)(1 − x4).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 13 / 34

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SLIDE 46

Generalizations

  • Szeg˝
  • also showed positivity of

(and extension to any # of variables)

1 4

i=1

  • j=i(1 − xj)

= 1 (1 − x2)(1 − x3)(1 − x4) + · · · + (1 − x1)(1 − x2)(1 − x3)

  • The Lewy–Askey problem asks for positivity of

1

  • 1i<j4

(1 − xi)(1 − xj) = 1 (1 − x1)(1 − x2) + · · · + (1 − x3)(1 − x4).

  • Non-negativity proved by a very general result of Scott–Sokal (’13):
  • 1

det ((1 − xi)Ai) is non-negative if Ai 0 are hermitian matrices.

  • For the Lewy–Askey problem:

A1 =

  • 1
  • ,

A2 =

  • 1
  • ,

A3 =

  • 1

1 1 1

  • ,

A4 =

  • 1

e−iπ/3 eiπ/3 1

  • .

Properties of Laurent coefficients of multivariate rational functions Armin Straub 13 / 34

slide-47
SLIDE 47

Generalizations

  • Szeg˝
  • also showed positivity of

(and extension to any # of variables)

1 4

i=1

  • j=i(1 − xj)

= 1 (1 − x2)(1 − x3)(1 − x4) + · · · + (1 − x1)(1 − x2)(1 − x3)

  • The Lewy–Askey problem asks for positivity of

1

  • 1i<j4

(1 − xi)(1 − xj) = 1 (1 − x1)(1 − x2) + · · · + (1 − x3)(1 − x4).

  • Non-negativity proved by a very general result of Scott–Sokal (’13):
  • 1

det ((1 − xi)Ai) is non-negative if Ai 0 are hermitian matrices.

  • For the Lewy–Askey problem:

A1 =

  • 1
  • ,

A2 =

  • 1
  • ,

A3 =

  • 1

1 1 1

  • ,

A4 =

  • 1

e−iπ/3 eiπ/3 1

  • .

er(1−x)−β in n variables positive iff β (n−r)/2 (or β = 0)?

Q

2 r n

With complete monotonicity of er(x)−β, this is a conjecture of Scott-Sokal (’13). Multivariate asymptotics?

Properties of Laurent coefficients of multivariate rational functions Armin Straub 13 / 34

slide-48
SLIDE 48

Preserving positivity

  • Positivity of the Askey–Gasper rational function

Askey–Gasper ’77 Koornwinder ’78 Ismail–Tamhankar ’79 Gillis–Reznick–Zeilberger ’83

1 1 − (x + y + z) + 4xyz

Properties of Laurent coefficients of multivariate rational functions Armin Straub 14 / 34

slide-49
SLIDE 49

Preserving positivity

  • Positivity of the Askey–Gasper rational function

Askey–Gasper ’77 Koornwinder ’78 Ismail–Tamhankar ’79 Gillis–Reznick–Zeilberger ’83

1 1 − (x + y + z) + 4xyz

implies positivity, for any ε > 0, of

1 1 − (x + y + z) + (4 − ε)xyz

Properties of Laurent coefficients of multivariate rational functions Armin Straub 14 / 34

slide-50
SLIDE 50

Preserving positivity

  • Positivity of the Askey–Gasper rational function

Askey–Gasper ’77 Koornwinder ’78 Ismail–Tamhankar ’79 Gillis–Reznick–Zeilberger ’83

1 (1 − (x + y + z) + 4xyz)β

implies positivity, for any ε > 0, of

for β ( √ 17 − 3)/2 ≈ 0.56

1 1 − (x + y + z) + (4 − ε)xyz

Properties of Laurent coefficients of multivariate rational functions Armin Straub 14 / 34

slide-51
SLIDE 51

Preserving positivity

  • Positivity of the Askey–Gasper rational function

Askey–Gasper ’77 Koornwinder ’78 Ismail–Tamhankar ’79 Gillis–Reznick–Zeilberger ’83

1 (1 − (x + y + z) + 4xyz)β

implies positivity, for any ε > 0, of

for β ( √ 17 − 3)/2 ≈ 0.56

1 1 − (x + y + z) + (4 − ε)xyz

  • If F(x1, . . . , xn) is positive, so is, for 0 p 1,

Gillis–Reznick–Zeilberger ’83 S ’08 Kauers–Zeilberger ’08

Tp(F) = F

  • px1

1−(1−p)x1 , . . . , pxn 1−(1−p)xn

  • (1 − (1 − p)x1) · · · (1 − (1 − p)xn).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 14 / 34

slide-52
SLIDE 52

Preserving positivity

  • Positivity of the Askey–Gasper rational function

Askey–Gasper ’77 Koornwinder ’78 Ismail–Tamhankar ’79 Gillis–Reznick–Zeilberger ’83

1 (1 − (x + y + z) + 4xyz)β

implies positivity, for any ε > 0, of

for β ( √ 17 − 3)/2 ≈ 0.56

1 1 − (x + y + z) + (4 − ε)xyz

  • If F(x1, . . . , xn) is positive, so is, for 0 p 1,

Gillis–Reznick–Zeilberger ’83 S ’08 Kauers–Zeilberger ’08

Tp(F) = F

  • px1

1−(1−p)x1 , . . . , pxn 1−(1−p)xn

  • (1 − (1 − p)x1) · · · (1 − (1 − p)xn).

T1/2 1 1 − (x + y + z) + 4xyz = 1 1 − (x + y + z) + 3

4(xy + yz + zx)

Hence, we can conclude positivity of Szeg˝

  • ’s function e2(1−x, 1−y, 1−z)−1.

EG

S ’08

Properties of Laurent coefficients of multivariate rational functions Armin Straub 14 / 34

slide-53
SLIDE 53

The case of three variables

ha,b(x, y, z) = 1 1 − (x + y + z) + a(xy + yz + zx) + bxyz

ha,b is positive ⇐ ⇒

   b < 6(1 − a) b 2 − 3a + 2(1 − a)3/2 a 1 CONJ

S ’08

−2 −1 1 5 10 15 20 a b

  • ha,b is positive in the

green region

S ’08

  • The conditions in the

conjecture are necessary for positivity

S–Zudilin ’15

Properties of Laurent coefficients of multivariate rational functions Armin Straub 15 / 34

slide-54
SLIDE 54

A conjecture of Gillis, Reznick and Zeilberger

For any d 4, the following function is non-negative: 1 1 − (x1 + x2 + . . . + xd) + d!x1x2 · · · xd

CONJ

G-R-Z ’83

Properties of Laurent coefficients of multivariate rational functions Armin Straub 16 / 34

slide-55
SLIDE 55

A conjecture of Gillis, Reznick and Zeilberger

For any d 4, the following function is non-negative: 1 1 − (x1 + x2 + . . . + xd) + d!x1x2 · · · xd

CONJ

G-R-Z ’83

Suffices to prove that the diagonal coefficients are non-negative.

THM

G-R-Z

Properties of Laurent coefficients of multivariate rational functions Armin Straub 16 / 34

slide-56
SLIDE 56

A conjecture of Gillis, Reznick and Zeilberger

For any d 4, the following function is non-negative: 1 1 − (x1 + x2 + . . . + xd) + d!x1x2 · · · xd

CONJ

G-R-Z ’83

Suffices to prove that the diagonal coefficients are non-negative.

THM

G-R-Z

“omitted due to its length”

proof

Properties of Laurent coefficients of multivariate rational functions Armin Straub 16 / 34

slide-57
SLIDE 57

A conjecture of Gillis, Reznick and Zeilberger

For any d 4, the following function is non-negative: 1 1 − (x1 + x2 + . . . + xd) + d!x1x2 · · · xd

CONJ

G-R-Z ’83

Suffices to prove that the diagonal coefficients are non-negative.

THM

G-R-Z

“omitted due to its length”

proof

  • False for d = 2, 3.
  • Kauers proved that diagonal is non-negative for d = 4, 5, 6.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 16 / 34

slide-58
SLIDE 58

A conjecture of Gillis, Reznick and Zeilberger

For any d 4, the following function is non-negative: 1 1 − (x1 + x2 + . . . + xd) + d!x1x2 · · · xd

CONJ

G-R-Z ’83

Suffices to prove that the diagonal coefficients are non-negative.

THM

G-R-Z

“omitted due to its length”

proof

  • False for d = 2, 3.
  • Kauers proved that diagonal is non-negative for d = 4, 5, 6.
  • With c in place of d!, the coefficient of x1 · · · xd is d! − c.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 16 / 34

slide-59
SLIDE 59

A conjecture of Gillis, Reznick and Zeilberger

For any d 4, the following function is non-negative: 1 1 − (x1 + x2 + . . . + xd) + d!x1x2 · · · xd

CONJ

G-R-Z ’83

Suffices to prove that the diagonal coefficients are non-negative.

THM

G-R-Z

“omitted due to its length”

proof

  • False for d = 2, 3.
  • Kauers proved that diagonal is non-negative for d = 4, 5, 6.
  • With c in place of d!, the coefficient of x1 · · · xd is d! − c.
  • Diagonal coefficients eventually positive if c < (d − 1)d−1?

Multivariate asymptotics?

Properties of Laurent coefficients of multivariate rational functions Armin Straub 16 / 34

slide-60
SLIDE 60

Positivity vs diagonal positivity

  • Consider rational functions F = 1/p(x1, . . . , xd) with p a symmetric

polynomial, linear in each variable. Under what condition(s) is the positivity of F implied by the positivity of its diagonal?

Q

Properties of Laurent coefficients of multivariate rational functions Armin Straub 17 / 34

slide-61
SLIDE 61

Positivity vs diagonal positivity

  • Consider rational functions F = 1/p(x1, . . . , xd) with p a symmetric

polynomial, linear in each variable. Under what condition(s) is the positivity of F implied by the positivity of its diagonal?

Q

1 1+x+y has positive diagonal coefficients but is not positive.

EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 17 / 34

slide-62
SLIDE 62

Positivity vs diagonal positivity

  • Consider rational functions F = 1/p(x1, . . . , xd) with p a symmetric

polynomial, linear in each variable. Under what condition(s) is the positivity of F implied by the positivity of its diagonal?

Q

1 1+x+y has positive diagonal coefficients but is not positive.

EG

F positive ⇐ ⇒ diagonal of F, and F|xd=0 are positive?

Q

SZ ’15

Properties of Laurent coefficients of multivariate rational functions Armin Straub 17 / 34

slide-63
SLIDE 63

Positivity vs diagonal positivity

  • Consider rational functions F = 1/p(x1, . . . , xd) with p a symmetric

polynomial, linear in each variable. Under what condition(s) is the positivity of F implied by the positivity of its diagonal?

Q

1 1+x+y has positive diagonal coefficients but is not positive.

EG

F positive ⇐ ⇒ diagonal of F, and F|xd=0 are positive?

Q

SZ ’15

F(x, y) = 1 1 + c1(x + y) + c2xy is positive ⇐ ⇒ diagonal of F, and F|y=0 are positive

THM

S-Zudilin 2015

  • d = 3: also yes, if the previous conjecture on ha,b is true.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 17 / 34

slide-64
SLIDE 64

Application: Szeg˝

  • ’s rational function, once more
  • Recall Szeg˝
  • ’s rational function

S(x, y, z) = 1 1 − (x + y + z) + 3

4(xy + yz + zx).

S(2x, 2y, 2z) has diagonal coefficients

sn =

n

  • k=0

(−27)n−k22k−n (3k)! k!3

  • k

n − k

  • ,

Properties of Laurent coefficients of multivariate rational functions Armin Straub 18 / 34

slide-65
SLIDE 65

Application: Szeg˝

  • ’s rational function, once more
  • Recall Szeg˝
  • ’s rational function

S(x, y, z) = 1 1 − (x + y + z) + 3

4(xy + yz + zx).

S(2x, 2y, 2z) has diagonal coefficients

sn =

n

  • k=0

(−27)n−k22k−n (3k)! k!3

  • k

n − k

  • ,

whose generating function is

y(z) = 2F1 1

3, 2 3

1

  • 27z(2 − 27z)
  • .

Properties of Laurent coefficients of multivariate rational functions Armin Straub 18 / 34

slide-66
SLIDE 66

Application: Szeg˝

  • ’s rational function, once more
  • Recall Szeg˝
  • ’s rational function

S(x, y, z) = 1 1 − (x + y + z) + 3

4(xy + yz + zx).

S(2x, 2y, 2z) has diagonal coefficients

sn =

n

  • k=0

(−27)n−k22k−n (3k)! k!3

  • k

n − k

  • ,

whose generating function is

y(z) = 2F1 1

3, 2 3

1

  • 27z(2 − 27z)
  • .
  • Ramanujan’s cubic transformation

2F1

  • 1

3, 2 3

1

  • 1 −

1 − x 1 + 2x 3 = (1 + 2x)2F1 1

3, 2 3

1

  • x3
  • ,

Properties of Laurent coefficients of multivariate rational functions Armin Straub 18 / 34

slide-67
SLIDE 67

Application: Szeg˝

  • ’s rational function, once more
  • Recall Szeg˝
  • ’s rational function

S(x, y, z) = 1 1 − (x + y + z) + 3

4(xy + yz + zx).

S(2x, 2y, 2z) has diagonal coefficients

sn =

n

  • k=0

(−27)n−k22k−n (3k)! k!3

  • k

n − k

  • ,

whose generating function is

y(z) = 2F1 1

3, 2 3

1

  • 27z(2 − 27z)
  • .
  • Ramanujan’s cubic transformation

2F1

  • 1

3, 2 3

1

  • 1 −

1 − x 1 + 2x 3 = (1 + 2x)2F1 1

3, 2 3

1

  • x3
  • ,

puts this in the form

y(z) = (1 + 2x(z))2F1 1

3, 2 3

1

  • x(z)3
  • ,

where the algebraic x(z) = c1z + c2z2 + . . . has positive coefficients.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 18 / 34

slide-68
SLIDE 68

Application: A conjecture of Kauers

The following rational function is positive: 1 1 − (x + y + z + w) + 64

27(yzw + xzw + xyw + xyz).

CONJ

Kauers 2007

  • The diagonal is positive.

S–Zudilin ’15

(apply CAD to recurrence of order 3 and degree 6)

  • The rational function obtained from setting w = 0 is positive.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 19 / 34

slide-69
SLIDE 69

Application: A conjecture of Kauers

The following rational function is positive: 1 1 − (x + y + z + w) + 64

27(yzw + xzw + xyw + xyz).

CONJ

Kauers 2007

  • The diagonal is positive.

S–Zudilin ’15

(apply CAD to recurrence of order 3 and degree 6)

  • The rational function obtained from setting w = 0 is positive.

(because 64/27 < 4)

Properties of Laurent coefficients of multivariate rational functions Armin Straub 19 / 34

slide-70
SLIDE 70

Application: Another conjecture of Kauers and Zeilberger

The following rational function is positive: 1 1 − (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw.

CONJ

Kauers- Zeilberger 2008

  • Would imply conjectured positivity of Lewy–Askey rational function

1 1 − (x + y + z + w) + 2

3(xy + xz + xw + yz + yw + zw).

Recent proof of non-negativity by Scott and Sokal, 2013

Properties of Laurent coefficients of multivariate rational functions Armin Straub 20 / 34

slide-71
SLIDE 71

Application: Another conjecture of Kauers and Zeilberger

The following rational function is positive: 1 1 − (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw.

CONJ

Kauers- Zeilberger 2008

  • Would imply conjectured positivity of Lewy–Askey rational function

1 1 − (x + y + z + w) + 2

3(xy + xz + xw + yz + yw + zw).

Recent proof of non-negativity by Scott and Sokal, 2013

The Kauers–Zeilberger function has diagonal coefficients dn =

n

  • k=0

n k 22k n 2 .

PROP

S-Zudilin 2015

Properties of Laurent coefficients of multivariate rational functions Armin Straub 20 / 34

slide-72
SLIDE 72

Arithmetically interesting diagonals

Remarkably, several further rational functions on the boundary of positivity have Ap´ ery-like diagonals:

1 1 − (x + y + z) + 4xyz

has diagonal coefficients

n

  • k=0

n k 3

.

EG

  • Next, time permitting: congruences stronger than Gauss for these

Properties of Laurent coefficients of multivariate rational functions Armin Straub 21 / 34

slide-73
SLIDE 73

Arithmetically interesting diagonals

Remarkably, several further rational functions on the boundary of positivity have Ap´ ery-like diagonals:

1 1 − (x + y + z) + 4xyz

has diagonal coefficients

n

  • k=0

n k 3

.

EG

Koornwinder’s rational function

1 1 − (x + y + z + w) + 4e3(x, y, z, w) − 16xyzw

has diagonal coefficients

n

  • k=0

2k k 22(n − k) n − k 2

.

Using a positivity preserving operator, implies positivity of

1/e3(1 − x, 1 − y, 1 − z, 1 − w)

EG

  • Next, time permitting: congruences stronger than Gauss for these

Properties of Laurent coefficients of multivariate rational functions Armin Straub 21 / 34

slide-74
SLIDE 74

III

Ap´ ery-like sequences

Properties of Laurent coefficients of multivariate rational functions Armin Straub 22 / 34

slide-75
SLIDE 75

Ap´ ery numbers and the irrationality of ζ(3)

  • The Ap´

ery numbers

1, 5, 73, 1445, . . .

A(n) =

n

  • k=0

n k 2n + k k 2 satisfy (n + 1)3A(n + 1) = (2n + 1)(17n2 + 17n + 5)A(n) − n3A(n − 1).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 23 / 34

slide-76
SLIDE 76

Ap´ ery numbers and the irrationality of ζ(3)

  • The Ap´

ery numbers

1, 5, 73, 1445, . . .

A(n) =

n

  • k=0

n k 2n + k k 2 satisfy (n + 1)3A(n + 1) = (2n + 1)(17n2 + 17n + 5)A(n) − n3A(n − 1). ζ(3) = ∞

n=1 1 n3 is irrational.

THM

Ap´ ery ’78

The same recurrence is satisfied by the “near”-integers B(n) =

n

  • k=0

n k 2n + k k 2  

n

  • j=1

1 j3 +

k

  • m=1

(−1)m−1 2m3n

m

n+m

m

 . Then, B(n)

A(n) → ζ(3). But too fast for ζ(3) to be rational.

proof

Properties of Laurent coefficients of multivariate rational functions Armin Straub 23 / 34

slide-77
SLIDE 77

Zagier’s search and Ap´ ery-like numbers

  • Recurrence for Ap´

ery numbers is the case (a, b, c) = (17, 5, 1) of (n + 1)3un+1 = (2n + 1)(an2 + an + b)un − cn3un−1. Are there other tuples (a, b, c) for which the solution defined by u−1 = 0, u0 = 1 is integral?

Q

Beukers, Zagier

Properties of Laurent coefficients of multivariate rational functions Armin Straub 24 / 34

slide-78
SLIDE 78

Zagier’s search and Ap´ ery-like numbers

  • Recurrence for Ap´

ery numbers is the case (a, b, c) = (17, 5, 1) of (n + 1)3un+1 = (2n + 1)(an2 + an + b)un − cn3un−1. Are there other tuples (a, b, c) for which the solution defined by u−1 = 0, u0 = 1 is integral?

Q

Beukers, Zagier

  • Essentially, only 14 tuples (a, b, c) found.

(Almkvist–Zudilin)

  • 4 hypergeometric and 4 Legendrian solutions (with generating functions

3F2

1

2, α, 1 − α

1, 1

  • 4Cαz
  • ,

1 1 − Cαz 2F1 α, 1 − α 1

  • −Cαz

1 − Cαz 2 ,

with α = 1

2, 1 3, 1 4, 1 6 and Cα = 24, 33, 26, 24 · 33)

  • 6 sporadic solutions
  • Similar (and intertwined) story for:
  • (n + 1)2un+1 = (an2 + an + b)un − cn2un−1

(Beukers, Zagier)

  • (n + 1)3un+1 = (2n + 1)(an2 + an + b)un − n(cn2 + d)un−1

(Cooper)

Properties of Laurent coefficients of multivariate rational functions Armin Straub 24 / 34

slide-79
SLIDE 79

The six sporadic Ap´ ery-like numbers

(a, b, c) A(n) (17, 5, 1)

Ap´ ery numbers

  • k

n k 2n + k n 2

(12, 4, 16)

  • k

n k 22k n 2

(10, 4, 64)

Domb numbers

  • k

n k 22k k 2(n − k) n − k

  • (7, 3, 81)

Almkvist–Zudilin numbers

  • k

(−1)k3n−3k n 3k n + k n (3k)! k!3

(11, 5, 125)

  • k

(−1)k n k 34n − 5k 3n

  • (9, 3, −27)
  • k,l

n k 2n l k l k + l n

  • Properties of Laurent coefficients of multivariate rational functions

Armin Straub 25 / 34

slide-80
SLIDE 80

Supercongruences for Ap´ ery numbers

  • Chowla, Cowles, Cowles (1980) conjectured that, for primes p 5,

A(p) ≡ 5 (mod p3).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 26 / 34

slide-81
SLIDE 81

Supercongruences for Ap´ ery numbers

  • Chowla, Cowles, Cowles (1980) conjectured that, for primes p 5,

A(p) ≡ 5 (mod p3).

  • Gessel (1982) proved that A(mp) ≡ A(m)

(mod p3).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 26 / 34

slide-82
SLIDE 82

Supercongruences for Ap´ ery numbers

  • Chowla, Cowles, Cowles (1980) conjectured that, for primes p 5,

A(p) ≡ 5 (mod p3).

  • Gessel (1982) proved that A(mp) ≡ A(m)

(mod p3). The Ap´ ery numbers satisfy the supercongruence

(p 5)

A(mpr) ≡ A(mpr−1) (mod p3r).

THM

Beukers, Coster ’85, ’88

Properties of Laurent coefficients of multivariate rational functions Armin Straub 26 / 34

slide-83
SLIDE 83

Supercongruences for Ap´ ery numbers

  • Chowla, Cowles, Cowles (1980) conjectured that, for primes p 5,

A(p) ≡ 5 (mod p3).

  • Gessel (1982) proved that A(mp) ≡ A(m)

(mod p3). The Ap´ ery numbers satisfy the supercongruence

(p 5)

A(mpr) ≡ A(mpr−1) (mod p3r).

THM

Beukers, Coster ’85, ’88

For primes p, simple combinatorics proves the congruence 2p p

  • =
  • k

p k

  • p

p − k

  • ≡ 1 + 1

(mod p2). For p 5, Wolstenholme’s congruence shows that, in fact, 2p p

  • ≡ 2

(mod p3).

EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 26 / 34

slide-84
SLIDE 84

Supercongruences for Ap´ ery numbers

  • Chowla, Cowles, Cowles (1980) conjectured that, for primes p 5,

A(p) ≡ 5 (mod p3).

  • Gessel (1982) proved that A(mp) ≡ A(m)

(mod p3). The Ap´ ery numbers satisfy the supercongruence

(p 5)

A(mpr) ≡ A(mpr−1) (mod p3r).

THM

Beukers, Coster ’85, ’88

Mathematica 7 miscomputes A(n) =

n

  • k=0

n k 2n + k k 2

for n > 5500.

A(5 · 113) = 12488301. . .about 2000 digits. . .about 8000 digits. . .795652125

Weirdly, with this wrong value, one still has

A(5 · 113) ≡ A(5 · 112) (mod 116).

EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 26 / 34

slide-85
SLIDE 85

Supercongruences for Ap´ ery-like numbers

  • Conjecturally, supercongruences like

A(mpr) ≡ A(mpr−1) (mod p3r) hold for all Ap´ ery-like numbers.

Osburn–Sahu ’09

  • Current state of affairs for the six sporadic sequences from earlier:

(a, b, c) A(n) (17, 5, 1)

  • k

n

k

2n+k

n

2

Beukers, Coster ’87-’88

(12, 4, 16)

  • k

n

k

22k

n

2

Osburn–Sahu–S ’16

(10, 4, 64)

  • k

n

k

22k

k

2(n−k)

n−k

  • Osburn–Sahu ’11

(7, 3, 81)

  • k(−1)k3n−3k n

3k

n+k

n

(3k)!

k!3

  • pen

modulo p3 Amdeberhan–Tauraso ’16

(11, 5, 125)

  • k(−1)kn

k

34n−5k

3n

  • Osburn–Sahu–S ’16

(9, 3, −27)

  • k,l

n

k

2n

l

k

l

k+l

n

  • pen

Robert Osburn Brundaban Sahu

(University of Dublin) (NISER, India) Properties of Laurent coefficients of multivariate rational functions Armin Straub 27 / 34

slide-86
SLIDE 86

Multivariate supercongruences

Define A(n) = A(n1, n2, n3, n4) by 1 (1 − x1 − x2)(1 − x3 − x4) − x1x2x3x4 =

  • n∈Z4

A(n)xn.

  • The Ap´

ery numbers are the diagonal coefficients.

  • For p 5, we have the multivariate supercongruences

A(npr) ≡ A(npr−1) (mod p3r).

THM

S 2014

Properties of Laurent coefficients of multivariate rational functions Armin Straub 28 / 34

slide-87
SLIDE 87

Multivariate supercongruences

Define A(n) = A(n1, n2, n3, n4) by 1 (1 − x1 − x2)(1 − x3 − x4) − x1x2x3x4 =

  • n∈Z4

A(n)xn.

  • The Ap´

ery numbers are the diagonal coefficients.

  • For p 5, we have the multivariate supercongruences

A(npr) ≡ A(npr−1) (mod p3r).

THM

S 2014

  • n0

a(n)xn = F(x) = ⇒

  • n0

a(pn)xpn = 1 p

p−1

  • k=0

F(ζk

p x) ζp = e2πi/p

  • Hence, both A(npr) and A(npr−1) have rational generating function.

The proof, however, relies on an explicit binomial sum for the coefficients.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 28 / 34

slide-88
SLIDE 88

Multivariate supercongruences

Define A(n) = A(n1, n2, n3, n4) by 1 (1 − x1 − x2)(1 − x3 − x4) − x1x2x3x4 =

  • n∈Z4

A(n)xn.

  • The Ap´

ery numbers are the diagonal coefficients.

  • For p 5, we have the multivariate supercongruences

A(npr) ≡ A(npr−1) (mod p3r).

THM

S 2014

  • By MacMahon’s Master Theorem,

A(n) =

  • k∈Z

n1 k n3 k n1 + n2 − k n1 n3 + n4 − k n3

  • .

Properties of Laurent coefficients of multivariate rational functions Armin Straub 28 / 34

slide-89
SLIDE 89

Multivariate supercongruences

Define A(n) = A(n1, n2, n3, n4) by 1 (1 − x1 − x2)(1 − x3 − x4) − x1x2x3x4 =

  • n∈Z4

A(n)xn.

  • The Ap´

ery numbers are the diagonal coefficients.

  • For p 5, we have the multivariate supercongruences

A(npr) ≡ A(npr−1) (mod p3r).

THM

S 2014

  • By MacMahon’s Master Theorem,

A(n) =

  • k∈Z

n1 k n3 k n1 + n2 − k n1 n3 + n4 − k n3

  • .
  • Because A(n − 1) = A(−n, −n, −n, −n), we also find

A(mpr − 1) ≡ A(mpr−1 − 1) (mod p3r).

Beukers ’85

Properties of Laurent coefficients of multivariate rational functions Armin Straub 28 / 34

slide-90
SLIDE 90

More conjectural multivariate supercongruences

  • Exhaustive search by Alin Bostan and Bruno Salvy:

1/(1 − p(x, y, z, w)) with p(x, y, z, w) a sum of distinct monomials; Ap´ ery numbers as diagonal

1 1 − (x + y + xy)(z + w + zw) 1 1 − (1 + w)(z + xy + yz + zx + xyz) 1 1 − (y + z + xy + xz + zw + xyw + xyzw) 1 1 − (y + z + xz + wz + xyw + xzw + xyzw) 1 1 − (z + xy + yz + xw + xyw + yzw + xyzw) 1 1 − (z + (x + y)(z + w) + xyz + xyzw)

Properties of Laurent coefficients of multivariate rational functions Armin Straub 29 / 34

slide-91
SLIDE 91

More conjectural multivariate supercongruences

  • Exhaustive search by Alin Bostan and Bruno Salvy:

1/(1 − p(x, y, z, w)) with p(x, y, z, w) a sum of distinct monomials; Ap´ ery numbers as diagonal

1 1 − (x + y + xy)(z + w + zw) 1 1 − (1 + w)(z + xy + yz + zx + xyz) 1 1 − (y + z + xy + xz + zw + xyw + xyzw) 1 1 − (y + z + xz + wz + xyw + xzw + xyzw) 1 1 − (z + xy + yz + xw + xyw + yzw + xyzw) 1 1 − (z + (x + y)(z + w) + xyz + xyzw)

The coefficients B(n) of each of these satisfy, for p 5, B(npr) ≡ B(npr−1) (mod p3r).

CONJ

S 2014

Properties of Laurent coefficients of multivariate rational functions Armin Straub 29 / 34

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SLIDE 92

An infinite family of rational functions

Let λ ∈ Zℓ

>0 with d = λ1 + . . . + λℓ. Define Aλ(n) by

1

  • 1jℓ
  • 1 −
  • 1rλj

xλ1+...+λj−1+r

  • − x1x2 · · · xd

=

  • n∈Zd

Aλ(n)xn.

  • If ℓ 2, then, for all primes p,

Aλ(npr) ≡ Aλ(npr−1) (mod p2r).

  • If ℓ 2 and max(λ1, . . . , λℓ) 2, then, for primes p 5,

Aλ(npr) ≡ Aλ(npr−1) (mod p3r).

THM

S 2014

λ = (2, 2) λ = (2, 1)

1 (1 − x1 − x2)(1 − x3 − x4) − x1x2x3x4 1 (1 − x1 − x2)(1 − x3) − x1x2x3 EG

Properties of Laurent coefficients of multivariate rational functions Armin Straub 30 / 34

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SLIDE 93

Further examples

1 (1 − x1 − x2)(1 − x3) − x1x2x3

has as diagonal the Ap´ ery-like numbers, associated with ζ(2),

B(n) =

n

  • k=0

n k 2n + k k

  • .

EG

1 (1 − x1)(1 − x2) · · · (1 − xd) − x1x2 · · · xd

has as diagonal the numbers

d = 3: Franel, d = 4: Yang–Zudilin

Yd(n) =

n

  • k=0

n k d .

EG

  • In each case, we obtain supercongruences generalizing results of

Coster (1988) and Chan–Cooper–Sica (2010).

Properties of Laurent coefficients of multivariate rational functions Armin Straub 31 / 34

slide-94
SLIDE 94

A conjectural multivariate supercongruence

The coefficients Z(n) of 1 1 − (x1 + x2 + x3 + x4) + 27x1x2x3x4 =

  • n∈Z4

Z(n)xn satisfy, for p 5, the multivariate supercongruences Z(npr) ≡ Z(npr−1) (mod p3r).

CONJ

S 2014

  • Here, the diagonal coefficients are the Almkvist–Zudilin numbers

Z(n) =

n

  • k=0

(−3)n−3k n 3k n + k n (3k)! k!3 ,

for which the univariate congruences are still open.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 32 / 34

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SLIDE 95

Some open problems

  • Which rational functions have the Gauss property?

A(npr) ≡ A(npr−1) (mod pr)

When are these necessarily combinations of x1···xm

f1···fm det

  • ∂fj

∂xi

  • ?
  • Which rational functions are positive?

When is diagonal, plus lower-dimensional, positivity sufficient?

  • Can we establish all supercongruences via rational functions?

1 1 − (x + y + z) + 4xyz , 1 1 − (x + y + z + w) + 27xyzw

  • Is there a rational function in three variables with the ζ(3)-Ap´

ery numbers as diagonal?

As Alin showed us, the GF is transcendental, so two variables is impossible.

Properties of Laurent coefficients of multivariate rational functions Armin Straub 33 / 34

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SLIDE 96

THANK YOU!

Slides for this talk will be available from my website: http://arminstraub.com/talks

  • F. Beukers, M. Houben, A. Straub

Gauss congruences for rational functions in several variables Preprint, 2017. arXiv:1710.00423

  • A. Straub, W. Zudilin

Positivity of rational functions and their diagonals Journal of Approximation Theory (special issue dedicated to Richard Askey), Vol. 195, 2015, p. 57-69

  • A. Straub

Multivariate Ap´ ery numbers and supercongruences of rational functions Algebra & Number Theory, Vol. 8, Nr. 8, 2014, p. 1985-2008

  • A. Straub

Positivity of Szeg¨

  • ’s rational function

Advances in Applied Mathematics, Vol. 41, Issue 2, Aug 2008, p. 255-264

Properties of Laurent coefficients of multivariate rational functions Armin Straub 34 / 34