h ∗ -polynomials of dilated lattice polytopes Katharina Jochemko KTH Stockholm Einstein Workshop Discrete Geometry and Topology, March 13, 2018
Lattice polytopes A set P ⊂ R d is a lattice polytope if there are x 1 , . . . , x m ∈ Z d with P = conv { x 1 , . . . , x m } .
Ehrhart theory The lattice point enumerator or discrete volume of P is � . � � P ∩ Z d � E( P ) := n = 1 n = 2 n = 3 E( nP ) = ( n + 1) 2 .
Ehrhart theory Theorem (Ehrhart’62) For every lattice polytope P in R d E P ( n ) := | nP ∩ Z d | agrees with a polynomial of degree dim P for n ≥ 1 . E P ( n ) is called the Ehrhart polynomial of P . Various combinatorial applications , i.e. ◮ posets (order preserving maps), ◮ graph colorings,... Central Questions ◮ Which polynomials are Ehrhart polynomials? ◮ Interpretation of coefficients ◮ roots, ...
Ehrhart series and h ∗ -polynomial Ehrhart series The Ehrhart series of an d -dimensional lattice polytope P ⊂ R d is defined by h ∗ 0 + h ∗ 1 t + · · · + h ∗ d t d � E P ( n ) t n = . (1 − t ) d +1 n ≥ 0 The numerator polynomial h ∗ P ( t ) is the h ∗ -polynomial of P . The vector h ∗ ( P ) := ( h ∗ 0 , . . . , h ∗ d ) is the h ∗ -vector .
Ehrhart series and h ∗ -polynomial Ehrhart series The Ehrhart series of an d -dimensional lattice polytope P ⊂ R d is defined by h ∗ 0 + h ∗ 1 t + · · · + h ∗ d t d � E P ( n ) t n = . (1 − t ) d +1 n ≥ 0 The numerator polynomial h ∗ P ( t ) is the h ∗ -polynomial of P . The vector h ∗ ( P ) := ( h ∗ 0 , . . . , h ∗ d ) is the h ∗ -vector . h ∗ -vector and coefficients of E P ( n ) Expansion into a binomial basis: � n + r � � n + r − 1 � � n � E P ( n ) = h ∗ + h ∗ + · · · + h ∗ . 0 1 d r r r
Inequalities for the h ∗ -vector Theorem (Stanley ’80) For every lattice polytope P in R d with h ∗ P = h ∗ 0 + h ∗ 1 t + · · · + h ∗ d t d h ∗ i ≥ 0 for all 0 ≤ i ≤ d. Question : Are there stronger inequalities for certain classes of polytopes? Such as... ◮ ...Unimodality : h ∗ 0 ≤ h ∗ 1 ≤ · · · ≤ h ∗ k ≥ · · · ≥ h ∗ d for some k ◮ ...Log-concavity : k ) 2 ≥ h ∗ ( h ∗ k − 1 h ∗ k +1 for all k ◮ ...Real-rootedness : h ∗ P = h ∗ 0 + h ∗ 1 t + · · · + h ∗ d t d has only real roots
IDP polytopes Conjecture (Stanley ’89) Every IDP polytope has a unimodal h ∗ -vector. A lattice polytope P ⊂ R d has the integer decomposition property (IDP) if for all integers n ≥ 1 and all p ∈ nP ∩ Z d p = p 1 + · · · + p n for some p 1 , . . . , p n ∈ P ∩ Z d . Examples ◮ unimodular simplex ◮ lattice parallelepiped ◮ lattice zonotope ◮ rP whenever r ≥ dim P − 1 (Bruns, Gubeladze, Trung ’97)
Dilated lattice polytopes Theorem (Brenti, Welker ’09; Diaconis, Fulman ’09; Beck, Stapledon ’10) Let P be a d-dimensional lattice polytope. Then there is an N such that the h ∗ -polynomial of rP has only real roots for r ≥ N. Conjecture (Beck, Stapledon ’10) Let P be a d-dimensional lattice polytope. Then the h ∗ -polynomial of rP has only real-roots whenever r ≥ d. Theorem (Higashitani ’14) Let P be a d-dimensional lattice polytope. Then the h ∗ -polynomial of rP has log-concave coefficients whenever r ≥ deg h ∗ P . Theorem (J. ’16) Let P be a d-dimensional lattice polytope. Then the h ∗ -polynomial of rP has only real roots whenever r ≥ deg h ∗ P .
Interlacing polynomials ◮ Proof of Kadison-Singer-Problem from 1959 (Marcus, Spielman, Srivastava ’15) ◮ Real-rootedness of independence polynomials of claw-free graphs (Chudnowski, Seymour ’07) compatible polynomials, common interlacers ◮ Real-rootedness of s -Eulerian polynomials (Savage, Visontai ’15) h ∗ -polynomial of s -Lecture hall polytopes are real-rooted Further literature: Br¨ anden ’14, Fisk ’08, Braun ’15
Interlacing polynomials
Interlacing polynomials Definition Let a , b , t 1 , . . . , t n , s 1 , . . . , s m ∈ R . Then f = a � m i =1 ( t − s i ) interlaces g = b � n i =1 ( t − t i ) and we write f � g if · · · ≤ s 2 ≤ t 2 ≤ s 1 ≤ t 1 Properties ◮ f � g if and only if cf � dg for all c , d � = 0. ◮ deg f ≤ deg g ≤ deg f + 1 ◮ α f + β g real-rooted for all α, β ∈ R
Interlacing polynomials
Polynomials with only nonpositive, real roots Lemma (Wagner ’00) Let f , g , h ∈ R [ t ] be real-rooted polynomials with only nonpositive, real roots and positive leading coefficients. Then (i) if f � h and g � h then f + g � h. (ii) if h � f and h � g then h � f + g. (iii) g � f if and only if f � tg.
Interlacing sequences of polynomials Definition A sequence f 1 , . . . , f m is called interlacing if f i � f j whenever i ≤ j . Lemma Let f 1 , . . . , f m be an interlacing polynomials with only nonnegative coefficients. Then c 1 f 1 + c 2 f 2 + · · · + c m f m is real-rooted for all c 1 , . . . , c m ≥ 0 .
Interlacing sequences of polynomials
Constructing interlacing sequences Proposition (Fisk ’08; Savage, Visontai ’15) Let f 1 , · · · , f m be a sequence of interlacing polynomials with only negative roots and positive leading coefficients. For all 1 ≤ l ≤ m let g l = tf 1 + · · · + tf l − 1 + f l + · · · + f m . Then also g 1 , · · · , g m are interlacing, have only negative roots and positive leading coefficients.
Linear operators preserving interlacing sequences Let F n + the collection of all interlacing sequences of polynomials with only nonnegative coefficients of length n . When does a matrix G = ( G i , j ( t )) ∈ R [ t ] m × n map F n + to F m + by G · ( f 1 , . . . , f n ) T ? Theorem (Br¨ and´ en ’15) Let G = ( G i , j ( t )) ∈ R [ t ] m × n . Then G : F n + → F m + if and only if (i) ( G i , j ( t )) has nonnegative entries for all i ∈ [ n ] , j ∈ [ m ] , and (ii) For all λ, µ > 0 , 1 ≤ i < j ≤ n, 1 ≤ k < l ≤ n ( λ t + µ ) G k , j ( t ) + G l , j ( t ) � ( λ t + µ ) G k , i ( t ) + G l , i ( t ) .
Example · · · 1 1 1 1 t 1 1 · · · 1 t t 1 · · · 1 ∈ R [ x ] ( n +1) × n . . . . . . . . . t t · · · t t (i) All entries have nonnegative coefficients � Submatrices: i j � � � � � � � � � � k G k , i ( t ) G k , j ( t ) 1 1 1 1 t 1 t t M = : l G l , i ( t ) G l , j ( t ) 1 1 t 1 t t t t (ii) ( λ t + µ ) G k , j ( t ) + G l , j ( t ) � ( λ t + µ ) G k , i ( t ) + G l , i ( t ) ( λ + 1) t + µ = ( λ t + µ ) · 1 + t � ( λ t + µ ) t + t = ( λ t + µ + 1) t �
Dilated lattice polytopes
Dilation operator For f ∈ R [[ t ]] and an integer r ≥ 1 there are uniquely determined f 0 , . . . , f r − 1 ∈ R [[ t ]] such that f ( t ) = f 0 ( t r ) + tf 1 ( t r ) + · · · + t r − 1 f r − 1 ( t r ) . For 0 ≤ i ≤ r − 1 we define f � r , i � = f i . Example: r = 2 1 + 3 t + 5 t 2 + 7 t 3 + t 5 Then f 1 = 3 + 7 t + t 2 f 0 = 1 + 5 t In particular, for all lattice polytopes P and all integers r ≥ 1 � r , 0 � E rP ( n ) t n = � � E P ( n ) t n n ≥ 0 n ≥ 0
h ∗ -polynomials of dilated polytopes Lemma (Beck, Stapledon ’10) Let P be a d-dimensional lattice polytope and r ≥ 1 . Then � � r , 0 � . h ∗ h ∗ P ( t )(1 + t + · · · + t r − 1 ) d +1 d � rP ( t ) = Equivalently, for h ∗ P =: h rP ( t ) = h � r , 0 � a � r , 0 � d +1 + h � r , 1 � ta � r , r − 1 � + · · · + h � r , r − 1 � ta � r , 1 � h ∗ d +1 , d +1 where (1 + t + · · · + t r − 1 ) d � � r , i � a � r , i � � ( t ) := d for all r ≥ 1 and all 0 ≤ i ≤ r − 1.
h ∗ rP ( t ) = (1 − t ) d +1 � E rP ( n ) t n n ≥ 0 � r , 0 � � = (1 − t ) d +1 E P ( n ) t n n ≥ 0 � r , 0 � (1 − t r ) d +1 � E P ( n ) t n = n ≥ 0 � r , 0 � (1 + t + · · · + t r − 1 ) d +1 (1 − t ) d +1 � E P ( n ) t n = n ≥ 0 � � r , 0 � (1 + t + · · · + t r − 1 ) d +1 h ∗ � = P ( t )
Another operator preserving interlacing... Proposition (Fisk ’08) Let f be a polynomial such that f � r , r − 1 � , . . . , f � r , 1 � , f � r , 0 � is an interlacing sequence. Let g ( t ) = (1 + t + · · · + t r − 1 ) f ( t ) . Then also g � r , r − 1 � , . . . , g � r , 1 � , g � r , 0 � is an interlacing sequence. Observation: · · · 1 1 1 1 g � r , r − 1 � f � r , r − 1 � t 1 1 · · · 1 . . . . t t 1 · · · 1 . . = . . . g � r , 1 � ... f � r , 1 � . . . . . . g � r , 0 � f � r , 0 � t t · · · t 1 Corollary The polynomials a � r , r − 1 � ( t ) , . . . , a � r , 1 � ( t ) , a � r , 0 � ( t ) form an interlacing d d d sequence of polynomials.
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