Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck - PowerPoint PPT Presentation

Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck (SF State) math.sfsu.edu/beck Joint with Alan Stapledon (MSRI & UBC) arXiv:0804.3639 to appear in Math. Zeitschrift

Warm-Up Trivia Let’s say we add two random 100-digit integers. How often should we ◮ expect to carry a digit? Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 2

Warm-Up Trivia Let’s say we add two random 100-digit integers. How often should we ◮ expect to carry a digit? How about if we add three random 100-digit integers? ◮ Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 2

Warm-Up Trivia Let’s say we add two random 100-digit integers. How often should we ◮ expect to carry a digit? How about if we add three random 100-digit integers? ◮ How about if we add four random 100-digit integers? ◮ Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 2

Warm-Up Trivia Let’s say we add two random 100-digit integers. How often should we ◮ expect to carry a digit? How about if we add three random 100-digit integers? ◮ How about if we add four random 100-digit integers? ◮ A d ( t ) m d t m = � The Eulerian polynomial A d ( t ) is defined through (1 − t ) d +1 m ≥ 0 Persi Diaconis will tell you that the coefficients of A d ( t ) (the Eulerian numbers) play a role here. . . Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 2

Ehrhart Polynomials P ⊂ R d – lattice polytope of dimension d (vertices in Z d ) Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3

Ehrhart Polynomials P ⊂ R d – lattice polytope of dimension d (vertices in Z d ) � m P ∩ Z d � L P ( m ) := # (discrete volume of P ) Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3

Ehrhart Polynomials P ⊂ R d – lattice polytope of dimension d (vertices in Z d ) � m P ∩ Z d � L P ( m ) := # (discrete volume of P ) � L P ( m ) t m Ehr P ( t ) := 1 + m ≥ 1 Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3

Ehrhart Polynomials P ⊂ R d – lattice polytope of dimension d (vertices in Z d ) � m P ∩ Z d � L P ( m ) := # (discrete volume of P ) � L P ( m ) t m Ehr P ( t ) := 1 + m ≥ 1 Theorem (Ehrhart 1962) L P ( m ) is a polynomial in m of degree d . Equivalently, h ( t ) Ehr P ( t ) = (1 − t ) d +1 where h ( t ) is a polynomial of degree at most d . Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3

Ehrhart Polynomials P ⊂ R d – lattice polytope of dimension d (vertices in Z d ) � m P ∩ Z d � L P ( m ) := # (discrete volume of P ) � L P ( m ) t m Ehr P ( t ) := 1 + m ≥ 1 Theorem (Ehrhart 1962) L P ( m ) is a polynomial in m of degree d . Equivalently, h ( t ) Ehr P ( t ) = (1 − t ) d +1 where h ( t ) is a polynomial of degree at most d . Write the Ehrhart h-vector of P as h ( t ) = h d t d + h d − 1 t d − 1 + · · · + h 0 , then d � m + d − j � � L P ( m ) = h j . d j =0 Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3

Ehrhart Polynomials P ⊂ R d – lattice polytope of dimension d (vertices in Z d ) � m P ∩ Z d � L P ( m ) := # (discrete volume of P ) h ( t ) L P ( m ) t m = � Ehr P ( t ) := 1 + (1 − t ) d +1 m ≥ 1 (Serious) Open Problem Classify Ehrhart h-vectors. Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 4

Ehrhart Polynomials P ⊂ R d – lattice polytope of dimension d (vertices in Z d ) � m P ∩ Z d � L P ( m ) := # (discrete volume of P ) h ( t ) L P ( m ) t m = � Ehr P ( t ) := 1 + (1 − t ) d +1 m ≥ 1 (Serious) Open Problem Classify Ehrhart h-vectors. L P ( nm ) t m as n increases. � Easier Problem Study Ehr n P ( t ) = 1 + m ≥ 1 Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 4

Why Should We Care? Linear systems are everywhere, and so polytopes are everywhere. ◮ Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5

Why Should We Care? Linear systems are everywhere, and so polytopes are everywhere. ◮ In applications, the volume of the polytope represented by a linear ◮ system measures some fundamental data of this system (“average”). Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5

Why Should We Care? Linear systems are everywhere, and so polytopes are everywhere. ◮ In applications, the volume of the polytope represented by a linear ◮ system measures some fundamental data of this system (“average”). Polytopes are basic geometric objects, yet even for these basic objects ◮ volume computation is hard and there remain many open problems. Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5

Why Should We Care? Linear systems are everywhere, and so polytopes are everywhere. ◮ In applications, the volume of the polytope represented by a linear ◮ system measures some fundamental data of this system (“average”). Polytopes are basic geometric objects, yet even for these basic objects ◮ volume computation is hard and there remain many open problems. Many discrete problems in various mathematical areas are linear ◮ problems, thus they ask for the discrete volume of a polytope in disguise. Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5

Why Should We Care? Linear systems are everywhere, and so polytopes are everywhere. ◮ In applications, the volume of the polytope represented by a linear ◮ system measures some fundamental data of this system (“average”). Polytopes are basic geometric objects, yet even for these basic objects ◮ volume computation is hard and there remain many open problems. Many discrete problems in various mathematical areas are linear ◮ problems, thus they ask for the discrete volume of a polytope in disguise. Polytopes are cool. ◮ Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5

General Properties of Ehrhart h-Vectors t m = h d t d + h d − 1 t d − 1 + · · · + h 0 � m P ∩ Z d � � Ehr P ( t ) = 1 + # (1 − t ) d +1 m ≥ 1 P ◦ ∩ Z d � � P ∩ Z d � � (Ehrhart) h 0 = 1 , h 1 = # − d − 1 , h d = # ◮ Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6

General Properties of Ehrhart h-Vectors t m = h d t d + h d − 1 t d − 1 + · · · + h 0 � m P ∩ Z d � � Ehr P ( t ) = 1 + # (1 − t ) d +1 m ≥ 1 P ◦ ∩ Z d � � P ∩ Z d � � (Ehrhart) h 0 = 1 , h 1 = # − d − 1 , h d = # ◮ (Ehrhart) vol P = 1 d ! ( h d + h d − 1 + · · · + h 1 + 1) ◮ Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6

General Properties of Ehrhart h-Vectors t m = h d t d + h d − 1 t d − 1 + · · · + h 0 � m P ∩ Z d � � Ehr P ( t ) = 1 + # (1 − t ) d +1 m ≥ 1 P ◦ ∩ Z d � � P ∩ Z d � � (Ehrhart) h 0 = 1 , h 1 = # − d − 1 , h d = # ◮ (Ehrhart) vol P = 1 d ! ( h d + h d − 1 + · · · + h 1 + 1) ◮ (Stanley 1980) h j ∈ Z ≥ 0 ◮ Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6

General Properties of Ehrhart h-Vectors t m = h d t d + h d − 1 t d − 1 + · · · + h 0 � m P ∩ Z d � � Ehr P ( t ) = 1 + # (1 − t ) d +1 m ≥ 1 P ◦ ∩ Z d � � P ∩ Z d � � (Ehrhart) h 0 = 1 , h 1 = # − d − 1 , h d = # ◮ (Ehrhart) vol P = 1 d ! ( h d + h d − 1 + · · · + h 1 + 1) ◮ (Stanley 1980) h j ∈ Z ≥ 0 ◮ (Stanley 1991) Whenever h s > 0 but h s +1 = · · · = h d = 0 , then ◮ h 0 + h 1 + · · · + h j ≤ h s + h s − 1 + · · · + h s − j for all 0 ≤ j ≤ s . Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6

General Properties of Ehrhart h-Vectors t m = h d t d + h d − 1 t d − 1 + · · · + h 0 � m P ∩ Z d � � Ehr P ( t ) = 1 + # (1 − t ) d +1 m ≥ 1 P ◦ ∩ Z d � � P ∩ Z d � � (Ehrhart) h 0 = 1 , h 1 = # − d − 1 , h d = # ◮ (Ehrhart) vol P = 1 d ! ( h d + h d − 1 + · · · + h 1 + 1) ◮ (Stanley 1980) h j ∈ Z ≥ 0 ◮ (Stanley 1991) Whenever h s > 0 but h s +1 = · · · = h d = 0 , then ◮ h 0 + h 1 + · · · + h j ≤ h s + h s − 1 + · · · + h s − j for all 0 ≤ j ≤ s . � d � (Hibi 1994) h 0 + · · · + h j +1 ≥ h d + · · · + h d − j for 0 ≤ j ≤ − 1 . ◮ 2 Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6

General Properties of Ehrhart h-Vectors t m = h d t d + h d − 1 t d − 1 + · · · + h 0 � m P ∩ Z d � � Ehr P ( t ) = 1 + # (1 − t ) d +1 m ≥ 1 P ◦ ∩ Z d � � P ∩ Z d � � (Ehrhart) h 0 = 1 , h 1 = # − d − 1 , h d = # ◮ (Ehrhart) vol P = 1 d ! ( h d + h d − 1 + · · · + h 1 + 1) ◮ (Stanley 1980) h j ∈ Z ≥ 0 ◮ (Stanley 1991) Whenever h s > 0 but h s +1 = · · · = h d = 0 , then ◮ h 0 + h 1 + · · · + h j ≤ h s + h s − 1 + · · · + h s − j for all 0 ≤ j ≤ s . � d � (Hibi 1994) h 0 + · · · + h j +1 ≥ h d + · · · + h d − j for 0 ≤ j ≤ − 1 . ◮ 2 (Hibi 1994) If h d > 0 then h 1 ≤ h j for 2 ≤ j < d . ◮ Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6

General Properties of Ehrhart h-Vectors t m = h d t d + h d − 1 t d − 1 + · · · + h 0 � m P ∩ Z d � � Ehr P ( t ) = 1 + # (1 − t ) d +1 m ≥ 1 (Stapledon 2009) Many more inequalities for the h j ’s arising from ◮ Kneser’s Theorem ( arXiv:0904.3035 ) Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 7