Classification of Combinatorial Polynomials (in particular, Ehrhart Polynomials of Zonotopes) Matthias Beck San Francisco State University Katharina Jochemko Kungliga Tekniska H¨ ogskolan Emily McCullough University of San Francisco
Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P ⊂ R d , � is a polynomial in t of degree d := � � t P ∩ Z d � ehr P ( t ) := dim P with leading coefficient vol P and constant term 1 . h ∗ ( z ) ehr P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 Equivalent descriptions of an Ehrhart polynomial: ehr P ( t ) = c d t d + c d − 1 t d − 1 + · · · + c 0 ◮ via roots of ehr P ( t ) ◮ � t + d � t + d − 1 � t ehr P ( t ) = h ∗ � + h ∗ � + · · · + h ∗ � Ehr P ( z ) − → ◮ 0 1 d d d d (Wide) Open Problem Classify Ehrhart polynomials. Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Two-dimensional Ehrhart Polynomials c 1 Essentially due to Pick (1899) and Scott (1976) (iii) (ii) (i) 1 c 2 1 Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P , ehr P ( t ) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1 . h ∗ ( z ) ehr P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 � t + d � t + d − 1 � t ehr P ( t ) = h ∗ � + h ∗ � + · · · + h ∗ � − → 0 1 d d d d Theorem (Macdonald 1971) ( − 1) d ehr P ( − t ) enumerates the interior lattice points in t P . Equivalently, � t + d − 1 � t + d − 2 � t − 1 � � � ehr P ◦ ( t ) = h ∗ + h ∗ + · · · + h ∗ 0 d − 1 d d d d Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P , ehr P ( t ) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1 . h ∗ ( z ) ehr P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 � t + d − 1 � t + d − 2 � t − 1 ehr P ◦ ( t ) = h ∗ � + h ∗ � + · · · + h ∗ � − → d − 1 0 d d d d Theorem (Stanley 1980) h ∗ 0 , h ∗ 1 , . . . , h ∗ d are nonnegative integers. d +1 − k > 0 then k P ◦ contains an integer point. Corollary If h ∗ Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Positivity Among Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P , ehr P ( t ) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1 . h ∗ ( z ) ehr P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 Theorem (Stanley 1980) h ∗ 0 , h ∗ 1 , . . . , h ∗ d are nonnegative integers. Theorem (Betke–McMullen 1985, Stapledon 2009) If h ∗ d > 0 then h ∗ ( z ) = a ( z ) + z b ( z ) where a ( z ) = z d a ( 1 z ) and b ( z ) = z d − 1 b ( 1 z ) with nonnegative coefficients. Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Positivity Among Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P , ehr P ( t ) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1 . h ∗ ( z ) ehr P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 Theorem (Stanley 1980) h ∗ 0 , h ∗ 1 , . . . , h ∗ d are nonnegative integers. Theorem (Betke–McMullen 1985, Stapledon 2009) If h ∗ d > 0 then h ∗ ( z ) = a ( z ) + z b ( z ) where a ( z ) = z d a ( 1 z ) and b ( z ) = z d − 1 b ( 1 z ) with nonnegative coefficients. Open Problem Try to prove the analogous theorem for your favorite combinatorial polynomial with nonnegative coefficients. Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Unimodality & Real-rooted Polynomials j =0 h j z j is unimodal if for some k ∈ { 0 , 1 , . . . , d } The polynomial h ( z ) = � d h 0 ≤ h 1 ≤ · · · ≤ h k ≥ · · · ≥ h d Crucial Example h ( z ) has only real roots Conjectures h ∗ ( z ) is unimodal/real-rooted for hypersimplices ◮ ◮ order polytopes alcoved polytopes ◮ lattice polytopes with unimodular triangulations ◮ IDP polytopes (integer decomposition property) ◮ Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Unimodality & Real-rooted Polynomials j =0 h j z j is unimodal if for some k ∈ { 0 , 1 , . . . , d } The polynomial h ( z ) = � d h 0 ≤ h 1 ≤ · · · ≤ h k ≥ · · · ≥ h d Crucial Example h ( z ) has only real roots Conjecture (Stanley 1989) h ∗ ( z ) is unimodal for IDP polytopes. Classic Example P = [0 , 1] d comes with the Eulerian polynomial h ∗ ( z ) Theorem (Schepers–Van Langenhoven 2013) h ∗ ( z ) is unimodal for lattice parallelepipeds. Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Zonotopes n � � The zonotope generated by v 1 , . . . , v n ∈ R d is � λ j v j : 0 ≤ λ j ≤ 1 j =1 h ∗ ( z ) is real rooted for lattice Theorem (M B–Jochemko–McCullough) zonotopes. Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Zonotopes n � � The zonotope generated by v 1 , . . . , v n ∈ R d is � λ j v j : 0 ≤ λ j ≤ 1 j =1 h ∗ ( z ) is real rooted for lattice Theorem (M B–Jochemko–McCullough) zonotopes. B–Jochemko–McCullough) The convex hull of the h ∗ -polyno- Theorem (M mials of all d -dimensional lattice zonotopes is the d -dimensional simplicial cone A 1 ( d + 1 , z ) + R ≥ 0 A 2 ( d + 1 , z ) + · · · + R ≥ 0 A d +1 ( d + 1 , z ) where we define an ( A, j ) -Eulerian polynomial as d − 1 � |{ σ ∈ S d : σ ( d ) = d + 1 − j and des( σ ) = k }| z k A j ( d, z ) := k =0 Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Eulerian Polynomials The (type A) Eulerian polynomials are d − 1 � |{ σ ∈ S d : des( σ ) = k }| z k A ( d, z ) := k =0 where des( σ ) is the number of descents σ ( j + 1) < σ ( j ) A ( d, z ) ( t + 1) d z t = � A ( d, z ) is symmetric, real rooted, and (1 − z ) d +1 t ≥ 0 Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Eulerian Polynomials The (type A) Eulerian polynomials are d − 1 � |{ σ ∈ S d : des( σ ) = k }| z k A ( d, z ) := k =0 where des( σ ) is the number of descents σ ( j + 1) < σ ( j ) A ( d, z ) ( t + 1) d z t = � A ( d, z ) is symmetric, real rooted, and (1 − z ) d +1 t ≥ 0 My favorite proof Compute the Ehrhart series of � � x ∈ R d : 0 ≤ x σ ( d ) ≤ x σ ( d − 1) ≤ · · · ≤ x σ (1) ≤ 1 [0 , 1] d = � x σ ( j +1) < x σ ( j ) if j ∈ Des( σ ) σ ∈ S d Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Eulerian Polynomials The (type A) Eulerian polynomials are d − 1 � |{ σ ∈ S d : des( σ ) = k }| z k A ( d, z ) := k =0 where des( σ ) is the number of descents σ ( j + 1) < σ ( j ) A ( d, z ) ( t + 1) d z t = � A ( d, z ) is symmetric, real rooted, and (1 − z ) d +1 t ≥ 0 d − 1 � |{ σ ∈ S d : σ ( d ) = d + 1 − j and des( σ ) = k }| z k A j ( d, z ) := k =0 seem to have first been used by Brenti–Welker (2008). They are not all symmetric but unimodal (Kubitzke–Nevo 2009) and real rooted (Savage– Visontai 2015). Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
The Geometry of Refined Eulerian Polynomials d − 1 � |{ σ ∈ S d : σ ( d ) = d + 1 − j and des( σ ) = k }| z k Lemma 1 A j ( d, z ) = k =0 is the h ∗ -polynomial of the half-open cube j := [0 , 1] d \ x ∈ R d : x d = x d − 1 = · · · = x d +1 − j = 1 C d � � Lemma 2 The h ∗ -polynomial of a half-open lattice parallelepiped is a linear combination of A j ( d, z ) . Lemma 3 p Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Zonotopal h ∗ -polynomials h ∗ ( z ) is real rooted for lattice Theorem (M B–Jochemko–McCullough) zonotopes. B–Jochemko–McCullough) The convex hull of the h ∗ -polyno- Theorem (M mials of all d -dimensional lattice zonotopes is the d -dimensional simplicial cone K := A 1 ( d + 1 , z ) + R ≥ 0 A 2 ( d + 1 , z ) + · · · + R ≥ 0 A d +1 ( d + 1 , z ) Open Problem Classify h ∗ -polynomials of d -dimensional lattice zonotopes. This is nontrivial: we can prove that each h ∗ -polynomial is actually in A 1 ( d + 1 , z ) + Z ≥ 0 A 2 ( d + 1 , z ) + · · · + Z ≥ 0 A d +1 ( d + 1 , z ) however, K is not IDP. (And the above is not complete either.) Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
Valuations A Z d -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, and ϕ ( P + x ) = ϕ ( P ) for all x ∈ Z d . Theorem (McMullen 1977) For any lattice polytope P ϕ ( t P ) z t = h ϕ 0 + h ϕ 1 z + · · · + h ϕ d ( P ) z d � (1 − z ) d +1 t ≥ 0 Ehrhart Polynomials of Zonotopes Matthias Beck, Katharina Jochemko & Emily McCullough
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