Ehrhart Polynomials of Zonotopes

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1 Matthias Beck San Francisco State University Katharina Jochemko Kungliga Tekniska Högskolan Emily McCullough University of San Francisco AMS Session Ehrhart Theory an its Applications Hunter College 2017

2 The purely formal language of geometry escribes aequately the reality of space. We might say, in this sense, that geometry is successful magic. I shoul like to state a converse: is not all magic, to the extent that it is successful, geometry? René Thom

3 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P R, ehr P (t) := tp Z is a polynomial in t of egree := im P with leaing coefficient vol P an constant term 1. Ehr P (z) := 1 + t 1 ehr P (t) z t = h (z) (1 z) +1 Equivalent escriptions of an Ehrhart polynomial: ehr P (t) = c t + c 1 t c 0 via roots of ehr P (t) Ehr P (z) ehr P (t) = h 0 ( t+ ) + h 1 ( t+ 1 ) + + h ( t ) (Wie) Open Problem Classify Ehrhart polynomials. Bil!

4 Two-imensional Ehrhart Polynomials c 1 Essentially ue to Pick (1899) an Scott (1976) (iii) (ii) 1 (i) 1 c 2

5 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, ehr P (t) is a polynomial in t of egree := im P with leaing coefficient vol P an constant term 1. Ehr P (z) := 1 + t 1 ehr P (t) z t = h (z) (1 z) +1 ehr P (t) = h 0 ( t+ ) + h 1 ( t+ 1 ) + + h ( t ) Theorem (Maconal 1971) ( 1) ehr P ( t) enumerates the interior lattice points in tp. Equivalently, ehr P (t) = h ( t+ 1 ) + h 1 ( t+ 2 ) + + h 0 ( t 1 )

6 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, ehr P (t) is a polynomial in t of egree := im P with leaing coefficient vol P an constant term 1. Ehr P (z) := 1 + t 1 ehr P (t) z t = h (z) (1 z) +1 ehr P (t) = h ( t+ 1 ) + h 1 ( t+ 2 ) + + h 0 ( t 1 ) Theorem (Stanley 1980) h 0, h 1,..., h are nonnegative integers. Corollary If h +1 k > 0 then kp contains an integer point.

7 Positivity Among Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, ehr P (t) is a polynomial in t of egree := im P with leaing coefficient vol P an constant term 1. Ehr P (z) := 1 + t 1 ehr P (t) z t = h (z) (1 z) +1 Theorem (Stanley 1980) h 0, h 1,..., h are nonnegative integers. Theorem (Betke McMullen 1985, Stapleon 2009) If h > 0 then h (z) = a(z) + z b(z) where a(z) = z a( 1 z ) an b(z) = z 1 b( 1 z ) with nonnegative coefficients.

8 Positivity Among Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, ehr P (t) is a polynomial in t of egree := im P with leaing coefficient vol P an constant term 1. Ehr P (z) := 1 + t 1 ehr P (t) z t = h (z) (1 z) +1 Theorem (Stanley 1980) h 0, h 1,..., h are nonnegative integers. Theorem (Betke McMullen 1985, Stapleon 2009) If h > 0 then h (z) = a(z) + z b(z) where a(z) = z a( 1 z ) an b(z) = z 1 b( 1 z ) with nonnegative coefficients. Open Problem Try to prove the analogous theorem for your favorite combinatorial polynomial with nonnegative coefficients.

9 Unimoality & Real-roote Polynomials The polynomial h(z) = j=0 h jz j is unimoal if for some k {0, 1,..., } h 0 h 1 h k h Crucial Example h(z) has only real roots Open Problem Classify lattice polytopes with unimoal/real-roote h (z)

10 Unimoality & Real-roote Polynomials The polynomial h(z) = j=0 h jz j is unimoal if for some k {0, 1,..., } h 0 h 1 h k h Crucial Example h(z) has only real roots Open Problem Classify lattice polytopes with unimoal/real-roote h (z) Conjecture (Stanley 1989) h (z) is unimoal for integrally close polytopes. Classic Example P = [0, 1] comes with the Eulerian polynomial h (z) Theorem (Schepers Van Langenhoven 2013) h (z) is unimoal for lattice parallelepipes.

11 Zonotopes The zonotope generate by v 1,..., v n R is { n j=1 } λ j v j : 0 λ j 1 Theorem (MB Jochemko McCullough) zonotopes. h (z) is real roote for lattice

12 Zonotopes The zonotope generate by v 1,..., v n R is { n j=1 } λ j v j : 0 λ j 1 Theorem (MB Jochemko McCullough) zonotopes. h (z) is real roote for lattice Theorem (MB Jochemko McCullough) The convex hull of the h -polynomials of all -imensional lattice zonotopes is the -imensional simplicial cone A 1 ( + 1, z) + R 0 A 2 ( + 1, z) + + R 0 A +1 ( + 1, z) where we efine an (A, j)-eulerian polynomial as A j (, z) := 1 k=0 {σ S : σ() = + 1 j an es(σ) = k} z k

13 Eulerian Polynomials The (type A) Eulerian polynomials are A(, z) := 1 k=0 {σ S : es(σ) = k} z k where es(σ) is the number of escents σ(j + 1) < σ(j) A(, z) is symmetric, real roote, an t 0 (t + 1) z t = A(, z) (1 z) +1

14 Eulerian Polynomials The (type A) Eulerian polynomials are A(, z) := 1 k=0 {σ S : es(σ) = k} z k where es(σ) is the number of escents σ(j + 1) < σ(j) A(, z) is symmetric, real roote, an t 0 (t + 1) z t = A(, z) (1 z) +1 My favorite proof Compute the Ehrhart series of [0, 1] = σ S { x R : 0 x σ() x σ( 1) x σ(1) 1 x σ(j+1) < x σ(j) if j Des(σ) }

15 Eulerian Polynomials The (type A) Eulerian polynomials are A(, z) := 1 k=0 {σ S : es(σ) = k} z k where es(σ) is the number of escents σ(j + 1) < σ(j) A(, z) is symmetric, real roote, an t 0 (t + 1) z t = A(, z) (1 z) +1 A j (, z) := 1 k=0 {σ S : σ() = + 1 j an es(σ) = k} z k seem to have first been use by Brenti Welker (2008). They are not all symmetric but unimoal (Kubitzke Nevo 2009) an real roote (Savage Visontai 2015).

16 The Geometry of Refine Eulerian Polynomials Lemma 1 A j (, z) = 1 k=0 is the h -polynomial of the half-open cube {σ S : σ() = + 1 j an es(σ) = k} z k C j := [0, 1] \ { x R : x = x 1 = = x +1 j = 1 } Lemma 2 The h -polynomial of a half-open lattice parallelepipe is a linear combination of A j (, z). Lemma 3 p

17 Zonotopal h -polynomials Theorem (MB Jochemko McCullough) zonotopes. h (z) is real roote for lattice Theorem (MB Jochemko McCullough) The convex hull of the h -polynomials of all -imensional lattice zonotopes is the -imensional simplicial cone K := A 1 ( + 1, z) + R 0 A 2 ( + 1, z) + + R 0 A +1 ( + 1, z) Open Problem Classify h -polynomials of -imensional lattice zonotopes. This is nontrivial: we can prove that each h -polynomial is actually in A 1 ( + 1, z) + Z 0 A 2 ( + 1, z) + + Z 0 A +1 ( + 1, z) however, K is not integrally close. (An the above is not complete either.)

18 Valuations A Z -valuation ϕ satisfies ϕ( ) = 0, ϕ(p Q) = ϕ(p) + ϕ(q) ϕ(p Q) whenever P, Q, P Q, P Q are lattice polytopes, an ϕ(p + x) = ϕ(p) for all x Z. Theorem (McMullen 1977) For any lattice polytope P ϕ(tp) z t t 0 = hϕ 0 + hϕ 1 z + + hϕ (P ) z (1 z) +1

19 Valuations A Z -valuation ϕ satisfies ϕ( ) = 0, ϕ(p Q) = ϕ(p) + ϕ(q) ϕ(p Q) whenever P, Q, P Q, P Q are lattice polytopes, an ϕ(p + x) = ϕ(p) for all x Z. Theorem (McMullen 1977) For any lattice polytope P ϕ(tp) z t t 0 = hϕ 0 + hϕ 1 z + + hϕ z (1 z) +1 Theorem (Jochemko Sanyal 2016) A Z -valuation ϕ satisfies h ϕ 0 for every lattice polytope if an only if ϕ( ) 0 for all lattice simplices. Theorem (MB Jochemko McCullough) h ϕ (z) is real roote for any lattice zonotope an any combinatorially positive valuation ϕ.

20 Matroi Connections Conjecture (Schepers Van Langenhoven 2013) An integrally close polytope with interior lattice points has an alternatingly increasing h -polynomial. Theorem (MB Jochemko McCullough) The Schepers Van Langenhoven Conjecture hols for coloop-free zonotopes (generate by v 1,..., v n none of which is containe in every maximally linearly inepenent subset). Main Iea The h -polynomial of a zonotope generate by the columns of a totally unimoular matrix can be expresse as a linear combination of A j (, z). This extens Stanley s 1980 theorem that for Z generate by V Z ehr Z (n) = g(i) n I where I ranges over all linearly inepenent subsets of V an g(i) enotes the greatest common ivisor of all maximal minors of the matrix with column vectors I. I

21 Type B Conjecture (Schepers Van Langenhoven 2013) An integrally close polytope with interior lattice points has an alternatingly increasing h -polynomial. Theorem (MB Jochemko McCullough) Conjecture hols for type-b zonotopes The Schepers Van Langenhoven } j=1 λ jv j : 1 λ j 1 { n Main tool Type-B Eulerian polynomials stemming from signe permutations (2t + 1) z t = t 0 Theorem (Brenti 1994) B(, z) is real roote. B(, z) (1 z) +1

22 Type B Conjecture (Schepers Van Langenhoven 2013) An integrally close polytope with interior lattice points has an alternatingly increasing h -polynomial. Theorem (MB Jochemko McCullough) Conjecture hols for type-b zonotopes The Schepers Van Langenhoven } j=1 λ jv j : 1 λ j 1 { n Main tool We efine the (B, l)-eulerian polynomials B l (, z) := {(σ, ɛ) B : ɛ σ() = + 1 l an es(σ, ɛ) = k} z k, k=0 prove that they are real roote an alternatingly increasing, an realize them as h -polynomials of half-open ±1-cubes.