Discrete Volume Computations for Polyhedra y
20
Matthias Beck San Francisco State University math.sfsu.edu/beck
Graduate Student Meeting Applied Algebra & Combinatorics
x 5 2 z Themes
Combinatorial polynomials Computation (complexity)
Generating functions Combinatorial structures
Polyhedra
Discrete Volume Computations for Polyhedra Matthias Beck 2 Themes
Nonlinear Algebra
Combinatorial polynomials Computation (complexity)
Generating functions Combinatorial structures
Polyhedra Linear Algebra
Discrete Volume Computations for Polyhedra Matthias Beck 2 Motivation I: Birkho↵–von Neumann Polytope
x11 x1n . ··· . n2 j xjk =1for all 1 k n Bn = . . R 0 : 80 1 2 k xjk =1for all 1 j n 9 < xn1 ... xnn P = @ A P : ;
Discrete Volume Computations for Polyhedra Matthias Beck 3 Motivation II: Polynomial Method 101
Theorem [Appel & Haken 1976] The chromatic number of any planar graph is at most 4.
This theorem had been a conjecture (conceived by Guthrie when trying to color maps) for 124 years.
Birkho↵[1912] says: Try polynomials!
[mathforum.org]
Four-Color Theorem Rephrased For a planar graph G,wehave G(4) > 0, that is, 4 is not a root of the polynomial G(k).
Discrete Volume Computations for Polyhedra Matthias Beck 4 Motivation II: Polynomial Method 101
Theorem [Appel & Haken 1976] The chromatic number of any planar graph is at most 4.
This theorem had been a conjecture (conceived by Guthrie when trying to color maps) for 124 years.
Birkho↵[1912] says: Try polynomials!
Stanley [EC 1] says: Try monomial algebras and generating functions! [mathforum.org]
Four-Color Theorem Rephrased For a planar graph G,wehave G(4) > 0, that is, 4 is not a root of the polynomial G(k).
Discrete Volume Computations for Polyhedra Matthias Beck 4 Discrete Volumes
Rational polyhedron Rd – solution set of a system of linear equalities & inequalities with integerP⇢ coe cients
Goal: understand Zd ... P\
(list) zm1zm2 zmd I 1 2 ··· d m Zd 2XP\
(count) Zd I P\
Discrete Volume Computations for Polyhedra Matthias Beck 5 Discrete Volumes
Rational polyhedron Rd – solution set of a system of linear equalities & inequalities with integerP⇢ coe cients
Goal: understand Zd ... P\
(list) zm1zm2 zmd I 1 2 ··· d m Zd 2XP\
(count) Zd I P\ 1 1 d I (volume) vol( )= lim Z P t td P\t !1
Discrete Volume Computations for Polyhedra Matthias Beck 5 Discrete Volumes
Rational polyhedron Rd – solution set of a system of linear equalities & inequalities with integerP⇢ coe cients
Goal: understand Zd ... P\
(list) zm1zm2 zmd I 1 2 ··· d m Zd 2XP\
(count) Zd I P\ 1 1 d I (volume) vol( )= lim Z P t td P\t !1
1 d d Ehrhart function L (t):= Z = t Z for t Z>0 P P\t P\ 2 Discrete Volume Computations for Polyhedra Matthias Beck 5 Why Polyhedra?
I Linear systems are everywhere, and so polyhedra are everywhere.
Discrete Volume Computations for Polyhedra Matthias Beck 6 Why Polyhedra?
I Linear systems are everywhere, and so polyhedra are everywhere.
I In applications, the volume of the polytope represented by a linear system measures some fundamental data of this system (“average”).
Discrete Volume Computations for Polyhedra Matthias Beck 6 Why Polyhedra?
I Linear systems are everywhere, and so polyhedra are everywhere.
I In applications, the volume of the polytope represented by a linear system measures some fundamental data of this system (“average”).
I Polytopes are basic geometric objects, yet even for these basic objects volume computation is hard and there remain many open problems.
Discrete Volume Computations for Polyhedra Matthias Beck 6 Why Polyhedra?
I Linear systems are everywhere, and so polyhedra are everywhere.
I In applications, the volume of the polytope represented by a linear system measures some fundamental data of this system (“average”).
I Polytopes are basic geometric objects, yet even for these basic objects volume computation is hard and there remain many open problems.
I Many discrete problems in various mathematical areas are linear problems, thus they ask for the discrete volume of a polytope in disguise.
Discrete Volume Computations for Polyhedra Matthias Beck 6 Why Polyhedra?
I Linear systems are everywhere, and so polyhedra are everywhere.
I In applications, the volume of the polytope represented by a linear system measures some fundamental data of this system (“average”).
I Polytopes are basic geometric objects, yet even for these basic objects volume computation is hard and there remain many open problems.
I Many discrete problems in various mathematical areas are linear problems, thus they ask for the discrete volume of a polytope in disguise.
I Much discrete geometry can be modeled using polynomials and, conversely, many combinatorial polynomials can be modeled geometrically.
Discrete Volume Computations for Polyhedra Matthias Beck 6 A Warm-Up Ehrhart Function
Lattice polytope Rd – convex hull of finitely points in Zd P⇢
d For t Z>0 let L (t):= t Z 2 P P\ Example 1:
= conv (0, 0), (1, 0), (0, 1) { } 2 = (x, y) R 0 : x + y 1 2
Discrete Volume Computations for Polyhedra Matthias Beck 7 A Warm-Up Ehrhart Function
Lattice polytope Rd – convex hull of finitely points in Zd P⇢
d For t Z>0 let L (t):= t Z 2 P P\ Example 1:
= conv (0, 0), (1, 0), (0, 1) { } 2 = (x, y) R 0 : x + y 1 2
Example 2:
2 =[0, 1]d (the unit cube in Rd)
Discrete Volume Computations for Polyhedra Matthias Beck 7 Ehrhart Polynomials
Theorem (Ehrhart 1962) For any lattice polytope , L (t) is a polynomial in t of degree dim with P leadingP coe cient vol and constant termP 1. P Equivalently, Ehr (z):=1+ L (t) zt is rational: P P t 1 X h (z) Ehr (z)= ⇤ P (1 z)dim +1 P where the Ehrhart h-vector h⇤(z) satisfies h⇤(0) = 1 and h⇤(1) = (dim )! vol( ). P P
Discrete Volume Computations for Polyhedra Matthias Beck 8 Ehrhart Polynomials
Theorem (Ehrhart 1962) For any lattice polytope , L (t) is a polynomial in t of degree dim with P leadingP coe cient vol and constant termP 1. P Equivalently, Ehr (z):=1+ L (t) zt is rational: P P t 1 X h (z) Ehr (z)= ⇤ P (1 z)dim +1 P where the Ehrhart h-vector h⇤(z) satisfies h⇤(0) = 1 and h⇤(1) = (dim )! vol( ). P P
1 Seeming dichotomy: vol( )= lim L (t) can be computed P t tdim P discretely via a finite amount of data.!1 P
Discrete Volume Computations for Polyhedra Matthias Beck 8 Ehrhart Polynomials
Theorem (Ehrhart 1962) For any lattice polytope , L (t) is a polynomial in t of degree d := dim withP leadingP coe cient vol and constant term 1. P P h (z) Ehr (z):=1+ L (t) zt = ⇤ P P (1 z)d+1 t 1 X
Equivalent descriptions of an Ehrhart polynomial:
d d 1 I L (t)=cd t + cd 1 t + + c0 P ··· I via roots of L (t) P t+d t+d 1 t I Ehr (z) L (t)=h0⇤ + h1⇤ + + hd⇤ P ! P d d ··· d
h⇤(z) is the binomial transform of L (t) P
Discrete Volume Computations for Polyhedra Matthias Beck 9 Ehrhart Polynomials
Theorem (Ehrhart 1962) For any lattice polytope , L (t) is a polynomial in t of degree d := dim withP leadingP coe cient vol and constant term 1. P P h (z) Ehr (z):=1+ L (t) zt = ⇤ P P (1 z)d+1 t 1 X
Equivalent descriptions of an Ehrhart polynomial:
d d 1 I L (t)=cd t + cd 1 t + + c0 P ··· I via roots of L (t) P t+d t+d 1 t I Ehr (z) L (t)=h0⇤ + h1⇤ + + hd⇤ P ! P d d ··· d Open Problem Classify Ehrhart polynomials.
Discrete Volume Computations for Polyhedra Matthias Beck 9 Two-dimensional Ehrhart Polynomials
c1
Essentially due to Pick (1899) and Scott (1976)
(iii) (ii)
1 (i)
1 c2
Discrete Volume Computations for Polyhedra Matthias Beck 10 Ehrhart Polynomials
Theorem (Ehrhart 1962) For any lattice polytope , L (t) is a polynomial in t of degree d := dim withP leadingP coe cient vol and constant term 1. P P h (z) Ehr (z):=1+ L (t) zt = ⇤ P P (1 z)d+1 t 1 X
t+d t+d 1 t L (t)=h0⇤ + h1⇤ + + hd⇤ ! P d d ··· d Theorem (Macdonald 1971) ( 1)dL ( t) enumerates the interior lattice points in t .Equivalently, P P t+d 1 t+d 2 t 1 L (t)=hd⇤ d + hd⇤ 1 d + + h0⇤ d P ···
Discrete Volume Computations for Polyhedra Matthias Beck 11 Ehrhart Polynomials
Theorem (Ehrhart 1962) For any lattice polytope , L (t) is a polynomial in t of degree d := dim withP leadingP coe cient vol and constant term 1. P P h (z) Ehr (z):=1+ L (t) zt = ⇤ P P (1 z)d+1 t 1 X
t+d 1 t+d 2 t 1 L (t)=hd⇤ d + hd⇤ 1 d + + h0⇤ d ! P ···
Theorem (Stanley 1980) h0⇤,h1⇤,...,hd⇤ are nonnegative integers.
Discrete Volume Computations for Polyhedra Matthias Beck 11 Ehrhart Polynomials
Theorem (Ehrhart 1962) For any lattice polytope , L (t) is a polynomial in t of degree d := dim withP leadingP coe cient vol and constant term 1. P P h (z) Ehr (z):=1+ L (t) zt = ⇤ P P (1 z)d+1 t 1 X
t+d 1 t+d 2 t 1 L (t)=hd⇤ d + hd⇤ 1 d + + h0⇤ d ! P ···
Theorem (Stanley 1980) h0⇤,h1⇤,...,hd⇤ are nonnegative integers.
Corollary If hd⇤+1 k > 0 then k contains an integer point. P
Discrete Volume Computations for Polyhedra Matthias Beck 11 Interlude: Graph Coloring a la Ehrhart
k (k)=2 ... K2 2 ✓ ◆ k + 1
K2 k + 1
x1 = x 2 (Blass–Sagan)
Discrete Volume Computations for Polyhedra Matthias Beck 12 Interlude: Graph Coloring a la Ehrhart
k (k)=2 ... K2 2 ✓ ◆ k + 1
K2 k + 1
x1 = x 2 (Blass–Sagan)
Nonequation Algebra
Discrete Volume Computations for Polyhedra Matthias Beck 12 Interlude: Graph Coloring a la Ehrhart
k (k)=2 ... K2 2 k + 1 ✓ ◆
K2 k + 1
x = x Similarly, for any given graph 1 2 (Blass–Sagan) G on d nodes, we can write
k + d k + d 1 k (k)= ⇤ + ⇤ + + ⇤ G 0 d 1 d ··· d d ✓ ◆ ✓ ◆ ✓ ◆ for some (meaningful) nonnegative integers 0⇤,..., d⇤
Discrete Volume Computations for Polyhedra Matthias Beck 12 Interlude: Graph Coloring a la Ehrhart
k (k)=2 ... K2 2 k + 1 ✓ ◆
K2 k + 1
x = x Similarly, for any given graph 1 2 (Blass–Sagan) G on d nodes, we can write
k + d k + d 1 k (k)= ⇤ + ⇤ + + ⇤ G 0 d 1 d ··· d d ✓ ◆ ✓ ◆ ✓ ◆
Half-Open Problem Prove that ⇤ > 0 for some 0 j 4 if G is planar. j
Discrete Volume Computations for Polyhedra Matthias Beck 12 Ehrhart h⇤ Positivity Refined
d t h⇤(z) Ehr (z):=1+ t Z z = P P\ (1 z)d+1 t 1 X
Theorem (Stanley 1980) h0⇤,h1⇤,...,hd⇤ are nonnegative integers.
Theorem (Betke–McMullen 1985, Stapledon 2009) If hd⇤ > 0 then
h⇤(z)=a(z)+zb(z)
d 1 d 1 1 where a(z)=z a(z) and b(z)=z b(z) with nonnegative coe cients.
Discrete Volume Computations for Polyhedra Matthias Beck 13 Ehrhart h⇤ Positivity Refined
d t h⇤(z) Ehr (z):=1+ t Z z = P P\ (1 z)d+1 t 1 X
Theorem (Stanley 1980) h0⇤,h1⇤,...,hd⇤ are nonnegative integers.
Theorem (Betke–McMullen 1985, Stapledon 2009) If hd⇤ > 0 then
h⇤(z)=a(z)+zb(z)
d 1 d 1 1 where a(z)=z a(z) and b(z)=z b(z) with nonnegative coe cients.
Open Problem Try to prove the analogous theorem for your favorite combinatorial polynomial with nonnegative coe cients.
Discrete Volume Computations for Polyhedra Matthias Beck 13 More Binomial Transforms
k + d k + d 1 k Chromatic polynomial (k)= ⇤ + ⇤ + + ⇤ G 0 d 1 d ··· d d ✓ ◆ ✓ ◆ ✓ ◆
d d 1 binomial transform G⇤ (z):= d⇤ z + d⇤ 1 z + + 0⇤ ! ··· Theorem (MB–Le´on2019+) Let G be a graph on d vertices. Then there d 1 d 1 1 exist symmetric polynomials aG(z)=z aG(z) and bG(z)=z bG(z) with positive integer coe cients such that
⇤ (z)=a (z) b (z) . G G G
Moreover, a0 a1 aj where 1 j d 1,andb0 b1 bj where 1 j d 2.
Discrete Volume Computations for Polyhedra Matthias Beck 14 More Binomial Transforms
k + d k + d 1 k Chromatic polynomial (k)= ⇤ + ⇤ + + ⇤ G 0 d 1 d ··· d d ✓ ◆ ✓ ◆ ✓ ◆
d d 1 binomial transform G⇤ (z):= d⇤ z + d⇤ 1 z + + 0⇤ ! ··· Theorem (MB–Le´on2019+) Let G be a graph on d vertices. Then there d 1 d 1 1 exist symmetric polynomials aG(z)=z aG(z) and bG(z)=z bG(z) with positive integer coe cients such that
⇤ (z)=a (z) b (z) . G G G
Moreover, a0 a1 aj where 1 j d 1,andb0 b1 bj where 1 j d 2. d 1 Theorem (Hersh–Swartz 2008) d⇤ j j⇤ for 2 j 2 Similar results hold for flow polynomials of graphs (Breuer–Dall 2011).
Discrete Volume Computations for Polyhedra Matthias Beck 14 Unimodal & Real-rooted Polynomials
d j The polynomial h⇤(z)= h⇤z is unimodal if for some k 0, 1,...,d j=0 j 2{ } P h⇤ h⇤ h⇤ h⇤ 0 1 ··· k ··· d
Crucial Example h⇤(z) has only real roots
Discrete Volume Computations for Polyhedra Matthias Beck 15 Unimodal & Real-rooted Polynomials
d j The polynomial h⇤(z)= h⇤z is unimodal if for some k 0, 1,...,d j=0 j 2{ } P h⇤ h⇤ h⇤ h⇤ 0 1 ··· k ··· d
Crucial Example h⇤(z) has only real roots
d Classic Example =[0, 1] comes with the Eulerian polynomial h⇤(z) P
Theorem (Schepers–Van Langenhoven 2013) h⇤(z) is unimodal for lattice parallelepipeds.
Theorem (MB–Jochemko–McCullough 2019) h⇤(z) is real rooted for lattice zonotopes.
Discrete Volume Computations for Polyhedra Matthias Beck 15 Unimodal & Real-rooted Polynomials
d j The polynomial h⇤(z)= h⇤z is unimodal if for some k 0, 1,...,d j=0 j 2{ } P h⇤ h⇤ h⇤ h⇤ 0 1 ··· k ··· d
Crucial Example h⇤(z) has only real roots
Conjectures h⇤(z) is unimodal/real-rooted for
I hypersimplices I order polytopes I alcoved polytopes
I lattice polytopes with unimodular triangulations
I IDP polytopes (integer decomposition property)
Discrete Volume Computations for Polyhedra Matthias Beck 15 A Polynomial Ansatz to Antimagic Graph Labelings
An antimagic labeling of G =(V,E) is an assignment E Z>0 such that ! each edge label 1, 2,..., E is used exactly once; I | | I the sum of the labels on all edges incident with a given node is unique.
Discrete Volume Computations for Polyhedra Matthias Beck 16 A Polynomial Ansatz to Antimagic Graph Labelings
An antimagic labeling of G =(V,E) is an assignment E Z>0 such that ! each edge label 1, 2,..., E is used exactly once; I | | I the sum of the labels on all edges incident with a given node is unique.
Conjecture [Hartsfield & Ringel 1990] Every connected graph except K2 has an antimagic labeling.
[Alon et al 2004] connected graphs with minimum degree c log V I | | I [B´erczi et al 2017] connected regular graphs I open for trees
Discrete Volume Computations for Polyhedra Matthias Beck 16 A Polynomial Ansatz to Antimagic Graph Labelings
An antimagic labeling of G =(V,E) is an assignment E Z>0 such that ! each edge label 1, 2,..., E is used exactly once; I | | I the sum of the labels on all edges incident with a given node is unique.
Idea Introduce a counting function: let AG⇤ (k) be the number of assignments of positive integers to the edges of G such that each edge label is in 1, 2,...,k and is distinct; I { } I the sum of the labels on all edges incident with a given node is unique.
Then G has an antimagic labeling if and only if A⇤ ( E ) > 0. G | |
Discrete Volume Computations for Polyhedra Matthias Beck 17 A Polynomial Ansatz to Antimagic Graph Labelings
An antimagic labeling of G =(V,E) is an assignment E Z>0 such that ! each edge label 1, 2,..., E is used exactly once; I | | I the sum of the labels on all edges incident with a given node is unique.
Idea Introduce a counting function: let AG⇤ (k) be the number of assignments of positive integers to the edges of G such that each edge label is in 1, 2,...,k and is distinct; I { } I the sum of the labels on all edges incident with a given node is unique.
Then G has an antimagic labeling if and only if A⇤ ( E ) > 0. G | |
Bad News The counting function AG⇤ (k) is in general not a polynomial:
4 22 3 2 38 0 if k is even, AC⇤ 4(k)=k 3 k + 17k 3 k + (2 if k is odd.
Discrete Volume Computations for Polyhedra Matthias Beck 17 A Polynomial Ansatz to Antimagic Graph Labelings
An antimagic labeling of G =(V,E) is an assignment E Z>0 such that ! each edge label 1, 2,..., E is used exactly once; I | | I the sum of the labels on all edges incident with a given node is unique.
New Idea Introduce another counting function: let AG(k) be the number of assignments of positive integers to the edges of G such that each edge label is in 1, 2,...,k ; I { } I the sum of the labels on all edges incident with a given node is unique.
Theorem (MB–Farahmand 2017) AG(k) is a quasipolynomial in k of period at most 2.IfG minus its loops is bipartite then AG(k) is a polynomial.
Corollary For bipartite graphs, A⇤ ( E ) > 0. G | |
Discrete Volume Computations for Polyhedra Matthias Beck 17 One Last Picture: Birkho↵–von Neumann Roots
For more about roots of (Ehrhart) polynomials, see Braun (2008) and Pfeifle (2010).
Discrete Volume Computations for Polyhedra Matthias Beck 18