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Discreet e Volume Computations \ for Polytopes: An Invitation to Ehrhart Theory Matthias Beck San Francisco State University math.sfsu.edu/beck

Meet my friends . . . If the solution set of a linear system of (in-)equalities is bounded, we call this solution set a polytope. Alternatively, a polytope is the convex hull of a finite set of points in R d . Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 2

Meet my friends . . . If the solution set of a linear system of (in-)equalities is bounded, we call this solution set a polytope. Alternatively, a polytope is the convex hull of a finite set of points in R d . Example: the 3-dimensional unit cube . . . ( x, y, z ) ∈ R 3 :       0 ≤ x ≤ 1   0 ≤ y ≤ 1     0 ≤ z ≤ 1   Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 2

Meet my friends . . . If the solution set of a linear system of (in-)equalities is bounded, we call this solution set a polytope. Alternatively, a polytope is the convex hull of a finite set of points in R d . Example: the 3-dimensional unit cube . . . . . . is the convex hull of ( x, y, z ) ∈ R 3 :   (0 , 0 , 0)     0 ≤ x ≤ 1   (1 , 0 , 0) 0 ≤ y ≤ 1 (0 , 1 , 0)     0 ≤ z ≤ 1   (0 , 0 , 1) (1 , 1 , 0) (1 , 0 , 1) (0 , 1 , 1) (1 , 1 , 1) Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 2

Meet my friends . . . The standard simplex x ∈ R d : x 1 + x 2 + · · · + x d ≤ 1 , x j ≥ 0 � � ∆ = = conv { (0 , 0 , . . . , 0) , (1 , 0 , 0 , . . . , 0) , (0 , 1 , 0 , . . . , 0) , . . . , (0 , 0 , . . . , 0 , 1) } Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 3

Meet my friends . . . The standard simplex x ∈ R d : x 1 + x 2 + · · · + x d ≤ 1 , x j ≥ 0 � � ∆ = = conv { (0 , 0 , . . . , 0) , (1 , 0 , 0 , . . . , 0) , (0 , 1 , 0 , . . . , 0) , . . . , (0 , 0 , . . . , 0 , 1) } The pyramid over the ( d − 1) -dimensional unit cube ✷ : the convex hull of ✷ (lifted into dimension d ) and (0 , 0 , . . . , 0 , 1) or ( x 1 , x 2 , . . . , x d ) ∈ R d : � � Pyr = 0 ≤ x 1 , x 2 , . . . , x d − 1 ≤ 1 − x d ≤ 1 Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 3

Meet my friends . . . The cross-polytope ( x 1 , x 2 , . . . , x d ) ∈ R d : | x 1 | + | x 2 | + · · · + | x d | ≤ 1 � � = ✸ = conv { ( ± 1 , 0 , . . . , 0) , (0 , ± 1 , 0 , . . . , 0) , . . . , (0 , . . . , 0 , ± 1) } Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 4

A Plug For Great, Free Software YOU should check out Ewgenij Gawrilow and Michael Joswig’s polymake www.math.tu-berlin.de/polymake Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 5

Today’s Goal Given a lattice polytope P (i.e., the extreme points are in Z d ), compute its (continuous) volume � vol P := d x . P Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 6

Today’s Goal Given a lattice polytope P (i.e., the extreme points are in Z d ), compute its (continuous) volume � vol P := d x . P Approach: Discretize the problem . . . P ∩ 1 � t Z d � # vol P = lim . t d t →∞ Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 6

Today’s Goal Given a lattice polytope P (i.e., the extreme points are in Z d ), compute its (continuous) volume � vol P := d x . P Approach: Discretize the problem . . . P ∩ 1 � t Z d � # vol P = lim . t d t →∞ For a positive integer t we define the discrete volume of P as P ∩ 1 t Z d � � L P ( t ) := # . Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 6

Today’s Goal Given a lattice polytope P (i.e., the extreme points are in Z d ), compute its (continuous) volume � vol P := d x . P Approach: Discretize the problem . . . P ∩ 1 � t Z d � # vol P = lim . t d t →∞ For a positive integer t we define the discrete volume of P as P ∩ 1 t Z d � � L P ( t ) := # . Today’s real goal: Given a lattice polytope P , compute L P ( t ) . Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 6

Why Should We Care? Linear systems are everywhere, and so polytopes are everywhere. ◮ Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 7

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”). Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 7

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. Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 7

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. Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 7

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. ◮ Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 7

A Warm-Up Example ( x, y ) ∈ R 2 : 0 ≤ x, y ≤ 1 � � Let’s consider the unit square ✷ = Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 8

A Warm-Up Example ( x, y ) ∈ R 2 : 0 ≤ x, y ≤ 1 � � Let’s consider the unit square ✷ = ✷ ∩ 1 t Z 2 � � L ✷ ( t ) = # = . . . Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 8

A Warm-Up Example ( x, y ) ∈ R 2 : 0 ≤ x, y ≤ 1 � � Let’s consider the unit square ✷ = ✷ ∩ 1 t Z 2 � = ( t + 1) 2 � L ✷ ( t ) = # Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 8

A Warm-Up Example ( x, y ) ∈ R 2 : 0 ≤ x, y ≤ 1 � � Let’s consider the unit square ✷ = ✷ ∩ 1 t Z 2 � = ( t + 1) 2 � L ✷ ( t ) = # t 2 + 2 t + 1 vol ( ✷ ) = lim = 1 t 2 t →∞ Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 8

A Warm-Up Example ( x, y ) ∈ R 2 : 0 ≤ x, y ≤ 1 � � Let’s consider the unit square ✷ = ✷ ∩ 1 t Z 2 � = ( t + 1) 2 � L ✷ ( t ) = # t 2 + 2 t + 1 vol ( ✷ ) = lim = 1 t 2 t →∞ ✷ ◦ = ( x, y ) ∈ R 2 : 0 < x, y < 1 � � Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 8

A Warm-Up Example ( x, y ) ∈ R 2 : 0 ≤ x, y ≤ 1 � � Let’s consider the unit square ✷ = ✷ ∩ 1 t Z 2 � = ( t + 1) 2 � L ✷ ( t ) = # t 2 + 2 t + 1 vol ( ✷ ) = lim = 1 t 2 t →∞ ✷ ◦ = ( x, y ) ∈ R 2 : 0 < x, y < 1 � � � ✷ ◦ ∩ 1 � = ( t − 1) 2 = t 2 − 2 t + 1 = L ✷ ( − t ) t Z 2 L ✷ ◦ ( t ) = # Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 8

A Warm-Up Example in General Dimension ( x 1 , x 2 , . . . , x d ) ∈ R d : 0 ≤ x j ≤ 1 � � For the unit d -cube ✷ = we obtain the analogous formulas L ✷ ( t ) = ( t + 1) d L ✷ ◦ ( t ) = ( t − 1) d . and Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 9

A Warm-Up Example in General Dimension ( x 1 , x 2 , . . . , x d ) ∈ R d : 0 ≤ x j ≤ 1 � � For the unit d -cube ✷ = we obtain the analogous formulas L ✷ ( t ) = ( t + 1) d L ✷ ◦ ( t ) = ( t − 1) d . and Note that d � d � � t k , L ✷ ( t ) = vol ( ✷ ) = 1 k k =0 := m ( m − 1)( m − 2) ··· ( m − n +1) � m � (where are the binomial coefficients) n ! n Discreet \ e Volume Computations for Polytopes: An Invitation to Ehrhart Theory Matthias Beck 9