Real-rootedness, Unimodality, and Symmetric
Decompositions of Polynomials
Liam Solus
KTH Royal Institute of Technology
solus@kth.se2 August 2018
Summer Workshop on Lattice Polytopes
Osaka UniversityOsaka, Japan

A Common Endeavour in Combinatorics:
Prove that a generating polynomial
p(z) = p0 + p1 z + · · · + pd z dpk ∈ Z≥0 , k ∈ {0, . . . , d}is unimodal; i.e.
p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd

for some t ∈ {0, . . . , d}.

Also: p(z) is symmetric with respect to m if pk = pm−k for all k ∈ {0, . . . , m}.

Two Common Approaches:(1) Decompose and use symmetry:
A product of symmetric and unimodal polynomials is symmetric and unimodal.
A sum of symmetric and unimodal polynomials with similar modes is unimodal.

(2) Factor and use roots:
If p(z) is real-rooted then it is also unimodal.

Ehrhart h∗ -polynomials:
P ⊂ Rn a lattice polytope of dimension d with(Ehrhart) h∗ -polynomial h∗ (P; z).
∆ = conv(v (0) , . . . , v (d) ) ⊂ Rn a lattice d-simplex
with open parallelpiped( d)XΠ◦∆ :=λi (v (i) , 1) ∈ Rn+1 : 0 < λi < 1 ,i=0and box polynomial (or local h∗ -polynomial):Xz xn+1 .
`∗ (∆; z) :=n+1(x1 ,...,xn+1 )∈Π◦∆ ∩ZTheorem (Betke and McMullen; 1985). Let T be a lattice triangulation of ad-dimensional lattice polytope P. ThenXh∗ (P; z) =h(linkT (∆); z)`∗ (∆; z),∆∈Twhere h(linkT (∆); z) is the h-polynomial of the link of ∆ in T .

A Corollary and the Question of Box Unimodality:
P is reflexive if h∗ (P; z) is symmetric with respect to d.
Corollary (Betke and McMullen; 1985). Let T be a lattice triangulation of theboundary of a d-dimensional reflexive polytope P. ThenXh∗ (P; z) =h(linkT (∆); z)`∗ (∆; z).
∆∈T`∗ (∆; z) is always symmetric with respect to dim(∆) + 1.
If T is regular then h(linkT (∆); z) is symmetric with respect tod − dim(∆) − 1 and unimodal.
Definition (Schepers and Van Langenhoven; 2013). A lattice triangulation ofP is box unimodal if it is regular and `∗ (∆; z) is unimodal for all ∆ ∈ T .
Reflexive polytopes with box unimodal triangulations have unimodalh∗ -polynomials.
Question (Schepers and Van Langenhoven; 2013). Which (IDP and reflexive)
lattice polytopes have box unimodal triangulations?

s-Lecture Hall Order Polytopes:s = (s1 , . . . , sn ) a sequence of positive integers.

s = (s1 , s2 , s3 ) = (2, 4, 3)

P := ([n], ) a naturally-labeled poset on
ground set [n].
The s-lecture hall simplex is
x1xnPns := x ∈ Rn : 0 ≤≤ ··· ≤≤1s1
sn
The order polytope isO(P) := {x ∈ Rn : 0 ≤ xi ≤ 1, xi ≤ xj if i  j}
The s-lecture hall order polytope isxixjnO(P, s) := x ∈ R : 0 ≤ xi ≤ si , ≤if i  j
sisj

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s-Lecture Hall Order Polytopes:(Stanley; 1986).O(P) admits a regulartriangulation into unimodular simplices,
called the canonical triangulation ofO(P).

(Brändén, Leander; 2016). O(P, s)admits a regular triangulation into s-lecturehall simplices, called the s-canonicaltriangulation of O(P, s).

Question. Is the s-canonical triangulation always box unimodal? First of all, is`∗ (Pns ; z) always unimodal?

Use the Roots!:
Theorem (Gustafsson, LS; 2018). The local h∗ -polynomial `∗ (Pns ; z) is always
real-rooted, and thus unimodal.
The Proof:`∗ (Pns ; z) admits a combinatorial interpretation using descent statistics ofinversion sequences.
Use the theory of interlacing polynomials.

If q interlaces p thenp + q is real-rooted!
(Use recursions!)

Corollary (Gustafsson, LS; 2018). The s-canonical triangulation of O(P; s) isbox unimodal. Moreover, if O(P, s) is reflexive then h∗ (O(P, s); z) is unimodal.

An Application:
Theorem (Brändén, Leander; 2016). Let P = ([n], ) be naturally-labeled,ranked poset with rank function ρ : [n] −→ Z≥0 , and let
si := ρ(i) + 1for all i ∈ [n].

∗Then h (O(P, s); z) is symmetric with respect to n − 1.

34

12

P=O(P, s) has h∗ -polynomial
1 + 9z + 9z 2 + z 3(s1 , s2 , s3 , s4 ) = (1, 1, 2, 2)
O(P, s)Conjecture (Brändén, Leander; 2016). h∗ (O(P, s); z) is unimodal for O(P, s)
as above.
Corollary (Gustafsson, LS; 2018). The conjecture is true if P has a uniqueminimal element.

A Second Corollary to Betke and McMullen:
For p(z) = p0 + p1 z + · · · + pd z d ∈ R[z], there exist unique polynomials a(z) andb(z) of degree at most d and d − 1, respectively, such thatp(z) = a(z) + zb(z),a(z) is symmetric with respect to d, andb(z) is symmetric with respect to d − 1.
Definition. We call the ordered pair of polynomials (a, b) the I -decompositionof p.
Corollary (Betke, McMullen; 1985). If P is a lattice polytope of dimension dcontaining an interior lattice point, then the polynomials in the I -decomposition ofh∗ (P; z) have only nonnegative coefficients.
If the polynomials in the I -decomposition of p(z) have only nonnegative
coefficients and are unimodal, then p(z) is alternatingly increasing; i.e.
p0 ≤ pd ≤ p1 ≤ pd−1 ≤ · · · ≤ pb d+1 c .
2(Beck, Jochemko, and McCullough; 2016) proved:
The h∗ -polynomial of any combinatorially positive valuation of a lattice
zonotope is real-rooted.
The h∗ -polynomial of any combinatorially positive valuation of a centrally
symmetric lattice zonotope is alternatingly increasing.
Question 1. For a generating polynomial p with I -decomposition (a, b), when area and b both real-rooted?
One Example (LS; 2017): The base r -simplices for numeral systems havereal-rooted a and b. Moreover, b interlaces a!
The base-r n-simplex is the convex hull
Br ,n = conv(e1 , . . . , en , −q) ⊂ Rn ,where q = ((r − 1)r i−1 )ni=1 ∈ Rn for r ∈ Z>1 .
Question 2. For a generating polynomial p with I -decomposition (a, b), does binterlace a?

One Answer to Question 1:
Theorem (Brändén, LS; 2018). Leti(t) =d
Xci t i (t + 1)d−i ,i=0

where ci ≥ 0 for 0 ≤ i ≤ d, andXn≥0i(t)z n =h(z).
(1 − z)d+1

If
c0 + c1 + · · · + ci ≤ cd + cd−1 + · · · + cd−ifor all 0 ≤ i ≤ d/2, then the I -decomposition of h(z) w.r.t. d is real-rooted.
Applications:The h∗ -polynomial for any combinatorially positive valuation of a lattice
zonotope with interior lattice points have real-rooted I -decompositions.
The I -decomposition of the h-polynomial of a barycentric subdivision of a
Cohen-Macaulay simplicial complex is real-rooted.
Moreover, each of these polynomials is alternatingly increasing.

An Answer to Question 2:
Theorem (Brändén, LS; 2018). Let p ∈ R[x] be degree d with I –decomposition(a, b) w.r.t. d, such that a and b have only nonnegative coefficients. If a and b
have degree d and d − 1, respectively, then the following are equivalent:
1b interlaces a,

2a interlaces p,

3b interlaces p,

4

The reciprocal of p interlaces p.

Applications:
The h∗ -polynomial for any combinatorially positive valuation of a centrally
symmetric lattice zonotope has I -decomposition (a, b) with b interlacing a.
The colored Eulerian polynomials and colored derangement polynomials
all have I -decomposition (a, b) with b interlacing a.
The latter answers a conjecture (Athanasiadis; 2017) on the real-rootednessof the local h-polynomial of the r th edgewise subdivision of the barycentric
subdivision of a simplex.

Thank You for Listening!

Some References:
P. Brändén and L. Solus. Symmetric Decompositions and Real-rootedness.
Preprint available soon!
N. Gustafsson and L. Solus. Derangements, Ehrhart Theory, and local
h-polynomials. arXiv number: 1807.05246.
L. Solus. Simplices for Numeral Systems. To appear in Transactions of theAmerican Mathematical Society (2017).