Real-rooted h -polynomials Katharina Jochemko TU Wien Einstein - PowerPoint PPT Presentation

Real-rooted h ∗ -polynomials Katharina Jochemko TU Wien Einstein Workshop on Lattice Polytopes, December 15, 2016

Unimodality and real-rootedness Let a 0 ,..., a d ≥ 0 be real numbers. Unimodality a 0 ≤ ⋯ ≤ a i ≥ ⋯ ≥ a d for some 0 ≤ i ≤ d

Unimodality and real-rootedness Let a 0 ,..., a d ≥ 0 be real numbers. Real-rootedness of a d t d + a d − 1 t d − 1 + ⋯ + a 1 t + a 0 ⇓ Log-concavity ≥ a i − 1 a i + 1 for all 0 ≤ i ≤ d a 2 i ⇓ Unimodality a 0 ≤ ⋯ ≤ a i ≥ ⋯ ≥ a d for some 0 ≤ i ≤ d

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

Lattice zonotopes Theorem (Schepers, Van Langenhoven ’13) The h ∗ -polynomial of any lattice parallelepiped is unimodal. Theorem (Beck, J., McCullough ’16) The h ∗ -polynomial of any lattice zonotope is real-rooted. Emily McCullough Matthias Beck

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 distinct 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 simple real roots whenever r ≥ max { deg h ∗ ( P ) + 1 , d } .

h ∗ -polynomials of IDP -polytopes Conjecture (Stanley 98; Hibi, Ohsugi ’06; Schepers, Van Langenhoven ’13) If P is IDP then the h ∗ -polynomial of P has unimodal coefficients. ▸ Parallelepipeds are IDP and zonotopes can be tiled by parallelepipeds Shephard ’74 ▸ For all r ≥ dim P − 1, rP is IDP (Bruns, Gubeladze, Trung ’97).

Outline Interlacing polynomials Lattice zonotopes Dilated lattice polytopes

Interlacing polynomials

Interlacing polynomials Definition A polynomial f = ∏ m i = 1 ( t − s i ) interlaces a polynomial g = ∏ n i = 1 ( t − t i ) and we write f ⪯ g if ⋯ ≤ s 2 ≤ t 2 ≤ s 1 ≤ t 1 Properties ▸ f and g are real-rooted ▸ 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 (schematical :-) )

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) + → F m Let G = ( G i , j ( t )) ∈ R [ t ] m × n . Then G ∶ F n + 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 ⎜ ⎟ ∈ R [ x ] ( n + 1 )× n ⎜ ⋯ ⎟ t t 1 1 ⎜ ⎟ ⎜ ⎟ ⋮ ⋮ ⋮ ⎝ ⎠ ⋯ t t t t (i) All entries have nonnegative coefficients Submatrices: i j G k , i ( t ) G k , j ( t ) ( G l , j ( t ) ) ( 1 1 ) ( 1 1 ) ( t t ) ( t t ) k 1 1 1 t M = ∶ G l , i ( t ) l 1 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

Lattice zonotopes

Eulerian polynomials We call i ∈ { 1 ,..., d − 1 } a descent of a permutation σ ∈ S d if σ ( i + 1 ) > σ ( i ) . The number of descents of σ is denoted by des σ and set a ( d , k ) = ∣{ σ ∈ S d ∶ des σ = k }∣ The Eulerian polynomial is d − 1 A ( d , t ) = a ( d , k ) t k ∑ k = 0 Example: ( 1 1 ) ∈ S 3 2 3 2 3 123 132 213 231 312 321 A ( 3 , t ) = 1 + 4 t + t 2 Theorem (Frobenius ’10) For all d ≥ 1 the Eulerian polynomial A ( d , t ) has only real roots.

h ∗ -polynomials For every lattice polytope P ⊂ R d let E P ( n ) = ∣ nP ∩ Z d ∣ be the Ehrhart polynomial of P . The h ∗ -polynomial h ∗ ( P )( t ) of P is defined by h ∗ ( P )( t ) E P ( n ) t n ∑ = ( 1 − t ) dim P + 1 . n ≥ 0 Half-open unimodular simplices For a unimodular d -simplex ∆ with facets F 1 ,..., F d + 1 E ∆ ( n ) = ( n + d ) ⇒ h ∗ ( ∆ )( t ) = 1 d More generally, for 0 ≤ i ≤ d k = 1 F k ( n ) = ( n + d − i ) ⇒ h ∗ ( ∆ )( t ) = t i E ∆ ∖⋃ i d

Unit cubes Partition of unit cube C d = [ 0 , 1 ] d { x ∈ C d ∶ x σ ( 1 ) ≤ x σ ( 2 ) ≤ ⋯ ≤ x σ ( d ) } C d = ⋃ σ ∈ S d x 2 1 id ( 12 ) x 1

Unit cubes Partition of unit cube C d = [ 0 , 1 ] d { x ∈ C d ∶ x σ ( 1 ) ≤ x σ ( 2 ) ≤ ⋯ ≤ x σ ( d ) , C d = ⊎ σ ∈ S d x σ ( i ) < x σ ( i + 1 ) , if i descent of σ } x 2 1 id ( 12 ) x 1

Unit cubes Partition of unit cube C d = [ 0 , 1 ] d { x ∈ C d ∶ x σ ( 1 ) ≤ x σ ( 2 ) ≤ ⋯ ≤ x σ ( d ) , C d = ⊎ σ ∈ S d x σ ( i ) < x σ ( i + 1 ) , if i descent of σ } x 2 1 id ( 12 ) x 1 h ∗ ( C d )( t ) = ∑ t des σ = A ( d , t ) σ ∈ S d

Refined Eulerian polynomials For every j ∈ [ d ] we define the j -Eulerian numbers a j ( d , k ) = ∣{ σ ∈ S d ∶ des σ = k ,σ ( 1 ) = j }∣ and the j -Eulerian polynomial A j ( d , k ) = d − 1 a j ( d , k ) t k ∑ k = 0 Example : d = 4 , j = 2 2134 2143 2314 2341 2413 2431 A ( 3 , t ) = 4 t + 2 t 2

Refined Eulerian polynomials Lemma (Brenti, Welker ’08) For all d ≥ 1 and all 1 ≤ j ≤ d + 1 A j ( d + 1 , t ) = ∑ tA k ( d , t ) + ∑ A k ( d , t ) . k < i k ≥ i Thus, A d + 1 = G ⋅ A d , where ⎛ ⎞ 1 1 1 1 ⋯ ⎜ ⎟ ⎜ ⎟ t 1 1 1 ⋯ A d = ( A 1 ( d , t ) ,..., A d ( d , t )) T ⎜ ⎟ ⎜ ⎟ and t t 1 1 ⋯ ⎜ ⎟ ⎝ ⎠ ⋮ ⋮ ⋮ t t t t ⋯ Theorem (Brenti, Welker ’08, Savage, Visontai ’15) For all 1 ≤ j ≤ d the j-Eulerian polynomial A j ( d , t ) is real-rooted.

Half-open unit cubes j = [ 0 , 1 ] d ∖ { x 1 = 0 ,..., x j = 0 } Partition of half-open unit cube C d { x ∈ C d j ∶ x σ ( 1 ) ≤ x σ ( 2 ) ≤ ⋯ ≤ x σ ( d ) , ⊎ = C d j σ ∈ S d x σ ( i ) < x σ ( i + 1 ) , if i descent of σ } x 2 1 id ( 12 ) x 1 ⎧ ⎪ if σ ( 1 ) ≤ j , ⎪ des σ + 1 h ∗ ( C d j )( t ) = ∑ σ ∈ S d t des j σ ⎨ des j σ = ⎪ where ⎪ ⎩ des σ otherwise.

Refined Eulerian numbers Claim : { σ ∈ S d ∶ des j σ = k } ≅ { σ ∈ S d + 1 ∶ des σ = k ,σ ( 1 ) = j + 1 } Proof by example: d = 5, j = 3 24351 ↦ 424351 ↦ 425361 Theorem (Beck, J., McCullough ’16) h ∗ ( C d j )( t ) = A j + 1 ( d + 1 , t ) .

Half-open parallelepipeds For v 1 ,..., v d ∈ Z d linear independent and I ⊆ [ d ] ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ I ( v 1 ,..., v d ) ∶ = ⎨ λ i v i ∶ 0 ≤ λ i ≤ 1 , 0 < λ i if i ∈ I ⎬ ∑ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ i ∈ [ d ] v 2 v 1 I = ∅ I = { 1 , 2 } I = { 1 }

Half-open parallelepipeds and zonotopes For K ⊆ [ d ] we denote b ( K ) = ∣ relint (◊({ v i } i ∈ K ) ∩ Z d ∣ Theorem (Beck, J., McCullough ’16+) h ∗ ( I ( v 1 ,..., v d ))( t ) = ∑ b ( K ) A ∣ I ∪ K ∣+ 1 ( d + 1 , t ) . K ⊆ [ d ] In particular, the h ∗ -vector of every half-open parallelepiped is real-rooted.

Zonotopes p Theorem (Beck, J., McCullough ’16) The h ∗ -polynomial of every lattice zonotope is real-rooted. Theorem (Beck, J., McCullough ’16) Let d ≥ 1 . Then the convex hull of the set of all h ∗ -polynomials of lattice zonotopes/parallelepipeds equals A 1 ( d + 1 , t ) + R ≥ 0 A 2 ( d + 1 , t ) + ⋯ + R ≥ 0 A d + 1 ( d + 1 , t ) .

Dilated lattice polytopes