Period collapse in Ehrhart quasi-polynomials Tyrrell B. McAllister 1 - - PowerPoint PPT Presentation

Period collapse in Ehrhart quasi-polynomials

Tyrrell B. McAllister1 Joint work with H´ el` ene Rochais1

1University of Wyoming

University of Kansas 22 May 2016

• T. B. McAllister (U. of Wyoming)

Ehrhart quasi-polynomials KU, 22 May 2016 1 / 22

Introduction — Polytopes: convex and non-convex

Definition

A convex rational polytope Q ⊆ Rn is the convex hull of a finite subset S ⊆ Qn. If S ⊆ Zn, then Q is integral. A rational polytope Q ⊆ Rn (not necessarily convex) is a topological ball that is a union Q =

• i∈I

Qi for some finite family {Qi : i ∈ I} of convex rational polytopes, all with the same affine span.

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Introduction — Counting integer points

Fix a rational polytope Q ⊆ Rn. Consider positive integer dilates:

1P 2P 4P

We are interested in the function counting the number of integer lattice points in the kth dilate: k → |kQ ∩ Zn|.

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Introduction — Examples

Let Q := [0, 1] ⊆ R. Then |kQ ∩ Z| = k + 1.

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Introduction — Examples

Let Q := [0, 1] ⊆ R. Then |kQ ∩ Z| = k + 1. Let Q := [0, 1

p] ⊆ R for p ∈ Z≥1. Then

|kQ ∩ Z| = k p

• + 1 = 1

pk − k p

• + 1 = 1

pk + k p

• ,

where

• k

p

• is the fractional part of k

p, and

• k

p

• = 1 −
• k

p

• .
• T. B. McAllister (U. of Wyoming)

Ehrhart quasi-polynomials KU, 22 May 2016 4 / 22

Introduction — Examples

Let Q := [0, 1] ⊆ R. Then |kQ ∩ Z| = k + 1. Let Q := [0, 1

p] ⊆ R for p ∈ Z≥1. Then

|kQ ∩ Z| = k p

• + 1 = 1

pk − k p

• + 1 = 1

pk + k p

• ,

where

• k

p

• is the fractional part of k

p, and

• k

p

• = 1 −
• k

p

• .

Let Q := [0, 1] × [0, 1

p] ⊆ R2. Then

• kQ ∩ Z2

= (k + 1) 1 pk + k p

• = 1

p k2 + 1 p + k p

• k +

k p

• .
• T. B. McAllister (U. of Wyoming)

Ehrhart quasi-polynomials KU, 22 May 2016 4 / 22

Introduction — Examples

Let Q := [0, 1] ⊆ R. Then |kQ ∩ Z| = k + 1. Let Q := [0, 1

p] ⊆ R for p ∈ Z≥1. Then

|kQ ∩ Z| = k p

• + 1 = 1

pk − k p

• + 1 = 1

pk + k p

• ,

where

• k

p

• is the fractional part of k

p, and

• k

p

• = 1 −
• k

p

• .

Let Q := [0, 1] × [0, 1

p] ⊆ R2. Then

• kQ ∩ Z2

= (k + 1) 1 pk + k p

• = 1

p k2 + 1 p + k p

• k +

k p

• .

Note: k →

• k

p

• is a periodic function Z → Q with period p.
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Introduction — Ehrhart’s theorem

Theorem (Ehrhart 1962)

Let ehrQ(k) := |kQ ∩ Zn| for k ∈ Z≥1. Then ehrQ(k) is a rational quasi-polynomial function of k. That is, ehrQ(k) = cQ

0 (k) + cQ 1 (k)k + cQ 2 (k)k2 + · · · + cQ n (k)kn

for some periodic functions cQ

i : Z → Q.

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Introduction — Ehrhart’s theorem

Theorem (Ehrhart 1962)

Let ehrQ(k) := |kQ ∩ Zn| for k ∈ Z≥1. Then ehrQ(k) is a rational quasi-polynomial function of k. That is, ehrQ(k) = cQ

0 (k) + cQ 1 (k)k + cQ 2 (k)k2 + · · · + cQ n (k)kn

for some periodic functions cQ

i : Z → Q.

Theorem (Ehrhart 1962 — Integral case)

Moreover, if Q is integral, then ehrQ(k) is a polynomial function of k: ehrQ(k) = cQ

0 + cQ 1 k + · · · + cQ n kn.

That is, all the cQ

i ’s are constant functions.

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Introduction — Our motivating question

Definition

In general, let Q be the ring of periodic functions Z → Q. Then Q[x] is the ring of quasi-polynomials. (Note: Q[x] contains Q[x] as a subring.)

Definition

Let Q ⊆ Rn be a rational polytope. Call ehrQ(x) ∈ Q[x] the Ehrhart quasi-polynomial of Q (or Ehrhart polynomial if ehrQ(x) ∈ Q[x]).

Motivating Question

Which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes? That is, when does f (x) ∈ Q[x] satisfy f (x) = ehrQ(x) for some rational polytope Q?

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Introduction — The polynomial case . . .

Many necessary conditions on Ehrhart polynomials of integral polytopes are known. Conditions on their coefficients: Stanley (1980, 1991); Betke & McMullen (1985); Hibi (1994), Haase & Nill & Payne (2009); Henk & Tagami (2009); Stapledon (2009). Conditions on their roots in C: Beck & De Loera & Develin & Pfeifle & Stanley (2005); Braun (2008); Pfeifle (2010).

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Introduction — The polynomial case . . .

Many necessary conditions on Ehrhart polynomials of integral polytopes are known. Conditions on their coefficients: Stanley (1980, 1991); Betke & McMullen (1985); Hibi (1994), Haase & Nill & Payne (2009); Henk & Tagami (2009); Stapledon (2009). Conditions on their roots in C: Beck & De Loera & Develin & Pfeifle & Stanley (2005); Braun (2008); Pfeifle (2010). Many nontrivial results! But still cannot characterize the Ehrhart polynomials of: convex integral polytopes in dimension > 2 (!), convex rational polytopes∗ in dimension > 1 (!!).

∗That is, pseudo-integral polytopes: rational polytopes Q such that ehrQ(x) is

a polynomial.

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Introduction — Recalibrating our question

Original Motivating Question (restated)

Which are the possible periodic functions ci : Z → Q that appear in ehrQ(x) =

n

• i=0

ci(x) xi, for rational polytopes Q ⊆ Rn? Humbled by the “merely” polynomial case, we adjust our goals:

New Motivating Question

Which are the possible periods of the functions ci : Z → Q that appear in ehrQ(x) =

n

• i=0

ci(x) xi, for rational polytopes Q ⊆ Rn?

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Introduction — Period sequences

Definition

Fix a rational polytope Q. Let pi ∈ Z≥1 be the period of the coefficient cQ

i

in ehrQ(x). The period sequence of Q is the tuple (p0, . . . , pn). Fact: If Q ⊆ Rn has non-empty interior, then the “leading coefficient” cQ

n

is a constant. Indeed, cQ

n = Vol(Q). Hence, pn = 1.

New Motivating Question (re-restated)

Which tuples (p0, . . . , pn−1, 1) ∈ (Z≥1)n are period sequences of rational polytopes?

• T. B. McAllister (U. of Wyoming)

Ehrhart quasi-polynomials KU, 22 May 2016 9 / 22

Introduction — Period sequences

Definition

Fix a rational polytope Q. Let pi ∈ Z≥1 be the period of the coefficient cQ

i

in ehrQ(x). The period sequence of Q is the tuple (p0, . . . , pn). Fact: If Q ⊆ Rn has non-empty interior, then the “leading coefficient” cQ

n

is a constant. Indeed, cQ

n = Vol(Q). Hence, pn = 1.

New Motivating Question (re-restated)

Which tuples (p0, . . . , pn−1, 1) ∈ (Z≥1)n are period sequences of rational polytopes?

Answer

In not-necessarily-convex case: All such tuples. In convex case: Open — but at least all of the form (p0, p1, 1, . . . , 1).

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Ehrhart quasi-polynomials KU, 22 May 2016 9 / 22

Main Results

Theorem (TM & H. Rochais, 2016+)

There exists a not-necessarily-convex rational polytope Q ⊆ Rn with period sequence (p0, . . . , pn−1, 1), for all p0, . . . , pn−1 ∈ Z≥1.

Theorem (TM & H. Rochais, 2016+)

There exists a convex rational polytope Q ⊆ Rn with period sequence (p0, p1, 1, . . . , 1), for all p0, p1 ∈ Z≥1.

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McMullen’s bound on coefficient periods — (Preliminaries)

Definition

A polytope is reticular iff its affine span contains a lattice point. Conv

• (−1

3 , 1 3), (1 3, 2 3)

• Reticular

Conv

• (−2

3 , 1 2), (2 3, 1 2)

• Not reticular

For every dimension i ≤ n, can ask: “Is every i-face reticular?”

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Ehrhart quasi-polynomials KU, 22 May 2016 11 / 22

McMullen’s bound on coefficient periods — (Preliminaries)

Definition

A polytope is reticular iff its affine span contains a lattice point. Conv

• (−1

3 , 1 3), (1 3, 2 3)

• Reticular

Conv

• (−2

3 , 1 2), (2 3, 1 2)

• Not reticular

For every dimension i ≤ n, can ask: “Is every i-face reticular?”

Definition

Fix polytope Q ⊆ Rn. For 0 ≤ i ≤ n, the ith McMullen index of Q is mi := min {k ∈ Z≥1 : every i-face of kP is reticular} .

Example

Polytope on left has m0 = 3 and m1 = 1. Polytope on right has m0 = 6 and m1 = 2.

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McMullen’s bound on coefficient periods

Remark

If Q ⊆ Rn has nonempty interior, then McMullen indices satisfy: mn = 1, m0 = denominator of Q = minimum k ∈ Z≥1 s.t. kQ is integral. mn | mn−1 | · · · | m0. In particular m0 ≥ · · · ≥ mn.

Theorem (McMullen 1978)

Let Q ⊆ Rn be a rational polytope. Let mi be the ith McMullen index of Q, and let pi be the period of cQ

i : Z → Q in

ehrQ(x) =

n

• i=0

cQ

i (x) xi.

Then pi ≤ mi.

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Period collapse

Definition

Period collapse occurs when pi < mi. Period collapse means that a high-denominator polytope “acts like” a low-denominator polytope with respect to lattice-point enumeration.

Examples of period collapse “in the wild”

Berenstein–Zelevinsky polytopes: Lattice points count multiplicity of Vν in the tensor product Vλ ⊗ Vµ of irreps of a semisimple Lie

• algebra. The denominator is arbitrarily large, but the periods are at

most 2 for types A, B, C, D (TM & De Loera 2006). Akhtar—Coates—Galkin—Kasprzyk “mutations”: Send a Fano polytope P to a Fano polytope Q while preserving the Hilbert series. Exhibits period collapse in Q∗ if P is reflexive while Q is not (ACGK 2012).

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Possible period sequences for polytopes

Theorem (M. Beck, S. Sam, & K. Woods 2007)

Given positive integers pn−1 | · · · | p0, the simplex Conv

• 1

p0 , 0, . . . , 0

• , . . . ,
• 0, . . . , 0,

1 pn−1

• , (0, . . . , 0)
• ⊆ Rn.

has period sequence (p0, . . . , pn−1, 1). Moreover, McMullen’s bounds are tight for this polytope. Hence, there exists a simplex in Rn with period sequence (p, 1, . . . , 1) for all p ∈ Z≥1. Our strategy: Construct polytopes with period sequence (1, . . . , 1, p, 1, . . . , 1). Then glue along lattice facets to get (p0, p1, . . . , pn−1, 1). Hence, need polytopes in which one of McMullen’s bounds is arbitrarily far from tight.

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Possible period sequences for polytopes

Note: Period sequence is invariant under addition of polynomials.

Definition

Quasi-polynomials f (x), g(x) ∈ Q[x] are equivalent if f (x) − g(x) ∈ Q[x]. Write f (x) ≡ g(x).

Con

Equivalence plays nicely with addition, but not with multiplication in

• general. (Q[x] ⊆

Q[x] is a subgroup w.r.t. +, not an ideal.)

Pros

If ehrP(x) ≡ ehrQ(x), then P and Q have the same period sequence. If Q is integral, then ehrQ(x) ≡ 0. If P ∩ Q is integral, then ehrP∪Q(x) ≡ ehrP(x) + ehrQ(x). Preserved when building height-1 pyramids! ehrP(x) ≡ ehrQ(x) = ⇒ ehr∆(P)(x) ≡ ehr∆(Q)(x)

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Building blocks

Fix p ∈ Z≥1 — the desired period of the ith coefficient. Let q := p(p − 1) + 1.

Theorem (TM & M. Moriarity 2016+)

Let ℓ := [− 1

p, 0] ⊆ R, and let P ⊆ R2 be the convex pentagon with

vertices u+, u−, v+, v−, w, where u± := ±qe1, v± := ±(q − 1)e1 + e2, w := q p e2. Then ehrP(x) ≡ − ehrℓ(x). That is, the “periodicity” of the coefficients in P and ℓ exactly cancel. How to see this?

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Warm-up: Building period sequence (1, p, 1)

R T3 T1 T2

1 p

1 − 1

p

The p = 2 case Have P = T1 ∪ T2 ∪ T3. Let R = ℓ × [−q, q].

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Warm-up: Building period sequence (1, p, 1)

R T3 T2 U1(T1)

1 p

1 − 1

p

The p = 2 case Have P = T1 ∪ T2 ∪ T3. Let R = ℓ × [−q, q]. Apply affine lattice automorphisms U1, U2 : Z2 → Z2. Get integral P′ := U1(T1) ∪ U2(T2) ∪ T3. So,

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Warm-up: Building period sequence (1, p, 1)

R T3 U1(T1) U2(T2)

1 p

1 − 1

p

The p = 2 case Have P = T1 ∪ T2 ∪ T3. Let R = ℓ × [−q, q]. Apply affine lattice automorphisms U1, U2 : Z2 → Z2. Get integral P′ := U1(T1) ∪ U2(T2) ∪ T3. So,

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Ehrhart quasi-polynomials KU, 22 May 2016 17 / 22

Warm-up: Building period sequence (1, p, 1)

R T3 U1(T1) U2(T2)

1 p

1 − 1

p

The p = 2 case Have P = T1 ∪ T2 ∪ T3. Let R = ℓ × [−q, q]. Apply affine lattice automorphisms U1, U2 : Z2 → Z2. Get integral P′ := U1(T1) ∪ U2(T2) ∪ T3. So, ehrP = ehrP′ + ehr(1/p, 1] ≡ ehr(1/p, 1] ≡ − ehrℓ .

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Warm-up: Building period sequence (1, p, 1)

R T3 U1(T1) U2(T2)

1 p

1 − 1

p

The p = 2 case Have P = T1 ∪ T2 ∪ T3. Let R = ℓ × [−q, q]. Apply affine lattice automorphisms U1, U2 : Z2 → Z2. Get integral P′ := U1(T1) ∪ U2(T2) ∪ T3. So, ehrP = ehrP′ + ehr(1/p, 1] ≡ ehr(1/p, 1] ≡ − ehrℓ . Thus, ehrR∪P ≡ (2qx + 1) ehrℓ + ehrP ≡ (2qx + 1) ehrℓ − ehrℓ = 2q x ehrℓ .

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Building a polytope with period sequence (1, p, 1, . . ., 1)

Recall: ℓ := [− 1

p, 0], and P ⊆ R2 is the pentagon above. Then

ehrP x ≡ − ehrℓ(x). Let Q ⊆ Rj. Construct the pyramid ∆(Q) ⊆ Rj+1 over Q by

1

embedding Q at height zero in Rj+1, yielding Q′ ⊆ Rj+1,

2

taking the convex hull with new standard-basis vector ∆(Q) := Conv(Q′ ∪ {ej+1})

Lemma

The equivalence above still holds after taking i-fold pyramids over P and ℓ: ehr∆i(P)(x) ≡ − ehr∆i(ℓ)(x)

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Building a polytope with period sequence (1, p, 1, . . ., 1)

By erecting pyramids over ℓ and P, we construct a convex polytope Q ⊆ Rn with period sequence (1, p, 1, . . . , 1). The 3-dimensional case, p = 2: By gluing with the BSW simplex along an integral facet, get convex polytope with period sequence (p0, p1, 1, . . . , 1).

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Cyclic polytopes

Question: What polytope in Ri with n + 1 vertices (0 ≤ i ≤ n) has the maximum number of facets possible? Answer (McMullen 1970, Stanley 1975): Cyclic polytopes (The Upper Bound Theorem). The moment curve: χi : R → Ri t → (t, t2, . . . , ti)

Definition

Fix T := {t0 < t1 < · · · < tn} ⊆ R. For 0 ≤ i ≤ n, the cyclic polytope Ci(T) ⊆ Ri is Ci(T) := Conv(χi(T)).

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Ehrhart polynomials of cyclic polytopes

In the Ehrhart polynomial of an integral polytope Q ⊆ Rn, the leading coefficient cQ

n and penultimate coefficient cQ n−1 have nice interpretations in

terms of volume. In general, the other coefficients have no such interpretations. But there is a beautiful exception!

Theorem (F. Liu 2005)

Fix T := {t0 < t1 < · · · < tn} ⊆ Z. Write Ci := Ci(T) ⊆ Ri. Then the Ehrhart polynomial of Cn ⊆ Rn is given by ehrCn(x) =

n

• i=0

Vol(Ci) xi Hence, ehrCn(x) = Vol(Cn) xn + ehrCn−1(x)

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Building a polytope with period sequence (p0, p1, . . . , pn−1, 1)

Recall: ℓ := [− 1

p, 0], P ⊆ R2 is the pentagon above, and Ci is the

i-dimensional cyclic polytope. Using Liu’s recurrence for ehrCn(x), we build a not-necessarily-convex polytope Q ⊆ Rn with period sequence (1, . . . , 1, pi, 1, . . . , 1) by gluing Ci × ∆n−i−1(ℓ) to Ci−1 × ∆n−i−1(P) along integral facets. By gluing these together, we get a now-aggressively-non-convex polytope with period sequence (p0, p1, . . . , pn−1, 1).

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