h -polynomials of dilated lattice polytopes

Transcription

1 h -polynomials of dilated lattice polytopes Katharina Jochemko KTH Stockholm Einstein Workshop Discrete Geometry and Topology, March 13, 2018

2 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 }.

3 Ehrhart theory The lattice point enumerator or discrete volume of P is E(P) := P Z d. n = 1 n = 2 n = 3 E(nP) = (n + 1) 2.

4 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,...

5 Ehrhart series and h -polynomial Ehrhart series The Ehrhart series of an d-dimensional lattice polytope P R d is defined by E P (n)t n = h 0 + h 1 t + + h d td (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.

6 Ehrhart series and h -polynomial Ehrhart series The Ehrhart series of an d-dimensional lattice polytope P R d is defined by E P (n)t n = h 0 + h 1 t + + h d td (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 E P (n) = h0 + h1 r r ) + + h d ( ) n. r

7 Inequalities for the h -vector Theorem (Stanley 80) For every lattice polytope P in R d with hp = h 0 + h 1 t + + h d td hi 0 for all 0 i d. Question: Are there stronger inequalities for certain classes of polytopes? Such as......unimodality: h0 h1 hk hd for some k...log-concavity: (hk) 2 hk 1h k+1 for all k...real-rootedness: h P = h 0 + h 1t + + h dt d has only real roots

8 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 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) p = p p n

9 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.

10 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: Bränden 14, Fisk 08, Braun 15

11 Interlacing polynomials

12 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 Properties s 2 t 2 s 1 t 1 f g if and only if cf dg for all c, d 0. deg f deg g deg f + 1 αf + βg real-rooted for all α, β R

13 Interlacing polynomials

14 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.

15 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 is real-rooted for all c 1,..., c m 0. c 1 f 1 + c 2 f c m f m

16 Interlacing sequences of polynomials

17 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 tf l 1 + f l + + f m. Then also g 1,, g m are interlacing, have only negative roots and positive leading coefficients.

18 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 (Brändén 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 (λt + µ)g k,j (t) + G l,j (t) (λt + µ)g k,i (t) + G l,i (t).

19 Example t t t t t t t R[x] (n+1) n (i) All entries have nonnegative coefficients Submatrices: ( i j ) k G M = k,i (t) G k,j (t) l G l,i (t) G l,j (t) : ( 1 ) 1 ( t 1 ) ( t 1 t t ) ( t t t t ) (ii) (λt + µ)g k,j (t) + G l,j (t) (λt + µ)g k,i (t) + G l,i (t) (λ + 1)t + µ = (λt + µ) 1 + t (λt + µ)t + t = (λt + µ + 1)t

20 Dilated lattice polytopes

21 Dilation operator For f R[[t]] and an integer r 1 there are uniquely determined f 0,..., f r 1 R[[t]] such that For 0 i r 1 we define Example: r = 2 Then f (t) = f 0 (t r ) + tf 1 (t r ) + + t r 1 f r 1 (t r ). f r,i = f i t + 5t 2 + 7t 3 + t 5 f 0 = 1 + 5t f 1 = 3 + 7t + t 2 In particular, for all lattice polytopes P and all integers r 1 E rp (n)t n = n 0 E P (n)t n n 0 r,0

22 h -polynomials of dilated polytopes Lemma (Beck, Stapledon 10) Let P be a d-dimensional lattice polytope and r 1. Then h rp(t) = ( h P(t)(1 + t + + t r 1 ) d+1 d ) r,0. Equivalently, for h P =: h where h rp(t) = h r,0 a r,0 d+1 + h r,1 ta r,r 1 d h r,r 1 ta r,1 d+1, for all r 1 and all 0 i r 1. a r,i d (t) := ( (1 + t + + t r 1 ) d) r,i

23 hrp(t) = (1 t) d+1 E rp (n)t n n 0 = (1 t) d+1 E P (n)t n n 0 = (1 t r ) d+1 E P (n)t n n 0 r,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) ) r,0

24 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: g r,r 1. g r,1 g r,0 = t t t t t t 1 f r,r 1. f r,1 f r,0 Corollary The polynomials a r,r 1 d (t),..., a r,1 d (t), a r,0 d (t) form an interlacing sequence of polynomials.

25 Putting the pieces together... 1) hrp (t) = h r,0 a r,0 d+1 + h r,1 ta r,r 1 d h r,r 1 ta r,1 d+1 2) a r,r 1 d+1 (t),..., a r,1 d+1 (t), a r,0 d+1 (t) interlacing a r,0 d+1 (t), ta r,r 1 d+1 (t),..., ta r,1 d+1 (t) interlacing Key observation: For r > deg h P (t) h r,i = h i 0 Theorem (J. 16) Let P be a d-dimensional lattice polytope. Then hrp (t) has only real roots whenever r deg hp (t).

26 Stapledon Decomposition

27 IDP polytopes with interior lattice points Question (Schepers, Van Langenhoven 13) For any IDP polytope P with interior lattice point, is the h -polynomial h P = d i=0 h i ti alternatingly increasing, i.e. h 0 h d h 1 h d 1? Observation alternatingly increasing unimodal with peak in the middle reflexive polytopes with regular unimodular triangulation lattice parallelepipeds (Schepers, Van Langenhoven 13) coloop-free lattice zonotopes (Beck, J., McCullough 16)

28 IDP polytopes with interior lattice points Question Is there a uniform bound N such that the h -polynomial of rp is alternatingly increasing for all r N? Codegree For any d-dimensional lattice polytope P with deg h P = s l := min{r 1: rp Z } = d + 1 s Theorem (Higashitani 14) The h -polynomial of rp is alternatingly increasing whenever r max{s, d + 1 s}.

29 Stapledon Decomposition Theorem (Stapledon 09) Let P be a lattice polytope with deg hp = s and codegree l = d + 1 s. Then (1 + t + + t l 1 )hp (t) can be uniquely decomposed as (1 + t + + t l 1 )h P(t) = a(t) + t l b(t), where a(t) = t d a( 1 t ) and b(t) = td l b( 1 t ) are palindromic polynomials with nonnegative coefficients. Consequences: a i 0 h 0 + h h i h d + h d h d i+1 (Hibi 90) b i 0 h s + h s h i h 0 + h h i (Stanley 91)

30 Stapledon Decomposition Observation Every polynomial h(t) of degree d can be uniquely decomposed into palindromic polynomials a(t) = t d a( 1 t ) and b(t) = td 1 b( 1 t ) such that h(t) = a(t) + tb(t). Proof : a 0 a 1 a 2 a 2 a 1 a 0 + b 0 b 1 b 2 b 1 b 0 h 0 h 1 h 2 h 3 h 4 h 5 Observation h(t) is alternatingly increasing a(t) and b(t) are unimodal

31 Stapledon Decomposition for dilated polytopes Theorem (J. 18+) Let P be a lattice polytope and for all r 1 let h rp(t) = a r (t) + tb r (t) be the unique decomposition into palindromic polynomials a r (t) = t d a r ( 1 t ) and b r (t) = t d 1 b r ( 1 t ). Then for all r d + 1. b r (t) a r (t)

32 Concluding remarks Bound for real-rootedness of hrp (t) is optimal for deg h (P)(t) d+1 2 (using result by Batyrev and Hofscheier 10) Crucial: Coefficients of h -polynomial are nonnegative. Other applications, e.g., Combinatorial positive valuations Hilbert series of Cohen-Macaulay domains

33 Concluding remarks Bound for real-rootedness of hrp (t) is optimal for deg h (P)(t) d+1 2 (using result by Batyrev and Hofscheier 10) Crucial: Coefficients of h -polynomial are nonnegative. Other applications, e.g., Combinatorial positive valuations Hilbert series of Cohen-Macaulay domains Katharina Jochemko: On the real-rootedness of the Veronese construction for rational formal power series, International Mathematics Research Notices (online first 2017). Thank you