Asymptotics of Ehrhart Series of Lattice...
Transcript of Asymptotics of Ehrhart Series of Lattice...
![Page 1: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/1.jpg)
Asymptotics of Ehrhart Seriesof Lattice Polytopes
Matthias Beck (SF State)
math.sfsu.edu/beck
Joint with Alan Stapledon (MSRI & UBC)
arXiv:0804.3639to appear in Math. Zeitschrift
![Page 2: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/2.jpg)
Warm-Up Trivia
I Let’s say we add two random 100-digit integers. How often should weexpect to carry a digit?
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 2
![Page 3: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/3.jpg)
Warm-Up Trivia
I Let’s say we add two random 100-digit integers. How often should weexpect to carry a digit?
I How about if we add three random 100-digit integers?
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 2
![Page 4: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/4.jpg)
Warm-Up Trivia
I Let’s say we add two random 100-digit integers. How often should weexpect to carry a digit?
I How about if we add three random 100-digit integers?
I How about if we add four random 100-digit integers?
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 2
![Page 5: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/5.jpg)
Warm-Up Trivia
I Let’s say we add two random 100-digit integers. How often should weexpect to carry a digit?
I How about if we add three random 100-digit integers?
I How about if we add four random 100-digit integers?
The Eulerian polynomial Ad(t) is defined through∑m≥0
md tm =Ad(t)
(1− t)d+1
Persi Diaconis will tell you that the coefficients of Ad(t) (the Euleriannumbers) play a role here. . .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 2
![Page 6: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/6.jpg)
Ehrhart Polynomials
P ⊂ Rd – lattice polytope of dimension d (vertices in Zd)
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3
![Page 7: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/7.jpg)
Ehrhart Polynomials
P ⊂ Rd – lattice polytope of dimension d (vertices in Zd)
LP(m) := #(mP ∩ Zd
)(discrete volume of P )
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3
![Page 8: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/8.jpg)
Ehrhart Polynomials
P ⊂ Rd – lattice polytope of dimension d (vertices in Zd)
LP(m) := #(mP ∩ Zd
)(discrete volume of P )
EhrP(t) := 1 +∑m≥1
LP(m) tm
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3
![Page 9: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/9.jpg)
Ehrhart Polynomials
P ⊂ Rd – lattice polytope of dimension d (vertices in Zd)
LP(m) := #(mP ∩ Zd
)(discrete volume of P )
EhrP(t) := 1 +∑m≥1
LP(m) tm
Theorem (Ehrhart 1962) LP(m) is a polynomial in m of degree d .Equivalently,
EhrP(t) =h(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
![Page 10: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/10.jpg)
Ehrhart Polynomials
P ⊂ Rd – lattice polytope of dimension d (vertices in Zd)
LP(m) := #(mP ∩ Zd
)(discrete volume of P )
EhrP(t) := 1 +∑m≥1
LP(m) tm
Theorem (Ehrhart 1962) LP(m) is a polynomial in m of degree d .Equivalently,
EhrP(t) =h(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) = hdtd +hd−1t
d−1 + · · ·+h0, then
LP(m) =d∑
j=0
hj
(m+ d− j
d
).
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 3
![Page 11: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/11.jpg)
Ehrhart Polynomials
P ⊂ Rd – lattice polytope of dimension d (vertices in Zd)
LP(m) := #(mP ∩ Zd
)(discrete volume of P )
EhrP(t) := 1 +∑m≥1
LP(m) tm =h(t)
(1− t)d+1
(Serious) Open Problem Classify Ehrhart h-vectors.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 4
![Page 12: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/12.jpg)
Ehrhart Polynomials
P ⊂ Rd – lattice polytope of dimension d (vertices in Zd)
LP(m) := #(mP ∩ Zd
)(discrete volume of P )
EhrP(t) := 1 +∑m≥1
LP(m) tm =h(t)
(1− t)d+1
(Serious) Open Problem Classify Ehrhart h-vectors.
Easier Problem Study EhrnP(t) = 1 +∑m≥1
LP(nm) tm as n increases.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 4
![Page 13: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/13.jpg)
Why Should We Care?
I Linear systems are everywhere, and so polytopes are everywhere.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5
![Page 14: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/14.jpg)
Why Should We Care?
I Linear systems are everywhere, and so polytopes are everywhere.
I In applications, the volume of the polytope represented by a linearsystem measures some fundamental data of this system (“average”).
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5
![Page 15: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/15.jpg)
Why Should We Care?
I Linear systems are everywhere, and so polytopes are everywhere.
I In applications, the volume of the polytope represented by a linearsystem measures some fundamental data of this system (“average”).
I Polytopes are basic geometric objects, yet even for these basic objectsvolume computation is hard and there remain many open problems.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5
![Page 16: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/16.jpg)
Why Should We Care?
I Linear systems are everywhere, and so polytopes are everywhere.
I In applications, the volume of the polytope represented by a linearsystem measures some fundamental data of this system (“average”).
I Polytopes are basic geometric objects, yet even for these basic objectsvolume computation is hard and there remain many open problems.
I Many discrete problems in various mathematical areas are linearproblems, thus they ask for the discrete volume of a polytope indisguise.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5
![Page 17: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/17.jpg)
Why Should We Care?
I Linear systems are everywhere, and so polytopes are everywhere.
I In applications, the volume of the polytope represented by a linearsystem measures some fundamental data of this system (“average”).
I Polytopes are basic geometric objects, yet even for these basic objectsvolume computation is hard and there remain many open problems.
I Many discrete problems in various mathematical areas are linearproblems, thus they ask for the discrete volume of a polytope indisguise.
I Polytopes are cool.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 5
![Page 18: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/18.jpg)
General Properties of Ehrhart h-Vectors
EhrP(t) = 1 +∑m≥1
#(mP ∩ Zd
)tm =
hdtd + hd−1t
d−1 + · · ·+ h0
(1− t)d+1
I (Ehrhart) h0 = 1 , h1 = #(P ∩ Zd
)− d− 1 , hd = #
(P◦ ∩ Zd
)
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6
![Page 19: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/19.jpg)
General Properties of Ehrhart h-Vectors
EhrP(t) = 1 +∑m≥1
#(mP ∩ Zd
)tm =
hdtd + hd−1t
d−1 + · · ·+ h0
(1− t)d+1
I (Ehrhart) h0 = 1 , h1 = #(P ∩ Zd
)− d− 1 , hd = #
(P◦ ∩ Zd
)I (Ehrhart) volP =
1d!
(hd + hd−1 + · · ·+ h1 + 1)
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6
![Page 20: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/20.jpg)
General Properties of Ehrhart h-Vectors
EhrP(t) = 1 +∑m≥1
#(mP ∩ Zd
)tm =
hdtd + hd−1t
d−1 + · · ·+ h0
(1− t)d+1
I (Ehrhart) h0 = 1 , h1 = #(P ∩ Zd
)− d− 1 , hd = #
(P◦ ∩ Zd
)I (Ehrhart) volP =
1d!
(hd + hd−1 + · · ·+ h1 + 1)
I (Stanley 1980) hj ∈ Z≥0
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6
![Page 21: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/21.jpg)
General Properties of Ehrhart h-Vectors
EhrP(t) = 1 +∑m≥1
#(mP ∩ Zd
)tm =
hdtd + hd−1t
d−1 + · · ·+ h0
(1− t)d+1
I (Ehrhart) h0 = 1 , h1 = #(P ∩ Zd
)− d− 1 , hd = #
(P◦ ∩ Zd
)I (Ehrhart) volP =
1d!
(hd + hd−1 + · · ·+ h1 + 1)
I (Stanley 1980) hj ∈ Z≥0
I (Stanley 1991) Whenever hs > 0 but hs+1 = · · · = hd = 0 , thenh0 + h1 + · · ·+ hj ≤ hs + hs−1 + · · ·+ hs−j for all 0 ≤ j ≤ s.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6
![Page 22: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/22.jpg)
General Properties of Ehrhart h-Vectors
EhrP(t) = 1 +∑m≥1
#(mP ∩ Zd
)tm =
hdtd + hd−1t
d−1 + · · ·+ h0
(1− t)d+1
I (Ehrhart) h0 = 1 , h1 = #(P ∩ Zd
)− d− 1 , hd = #
(P◦ ∩ Zd
)I (Ehrhart) volP =
1d!
(hd + hd−1 + · · ·+ h1 + 1)
I (Stanley 1980) hj ∈ Z≥0
I (Stanley 1991) Whenever hs > 0 but hs+1 = · · · = hd = 0 , thenh0 + h1 + · · ·+ hj ≤ hs + hs−1 + · · ·+ hs−j for all 0 ≤ j ≤ s.
I (Hibi 1994) h0 + · · ·+ hj+1 ≥ hd + · · ·+ hd−j for 0 ≤ j ≤⌊
d2
⌋− 1.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6
![Page 23: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/23.jpg)
General Properties of Ehrhart h-Vectors
EhrP(t) = 1 +∑m≥1
#(mP ∩ Zd
)tm =
hdtd + hd−1t
d−1 + · · ·+ h0
(1− t)d+1
I (Ehrhart) h0 = 1 , h1 = #(P ∩ Zd
)− d− 1 , hd = #
(P◦ ∩ Zd
)I (Ehrhart) volP =
1d!
(hd + hd−1 + · · ·+ h1 + 1)
I (Stanley 1980) hj ∈ Z≥0
I (Stanley 1991) Whenever hs > 0 but hs+1 = · · · = hd = 0 , thenh0 + h1 + · · ·+ hj ≤ hs + hs−1 + · · ·+ hs−j for all 0 ≤ j ≤ s.
I (Hibi 1994) h0 + · · ·+ hj+1 ≥ hd + · · ·+ hd−j for 0 ≤ j ≤⌊
d2
⌋− 1.
I (Hibi 1994) If hd > 0 then h1 ≤ hj for 2 ≤ j < d.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 6
![Page 24: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/24.jpg)
General Properties of Ehrhart h-Vectors
EhrP(t) = 1 +∑m≥1
#(mP ∩ Zd
)tm =
hdtd + hd−1t
d−1 + · · ·+ h0
(1− t)d+1
I (Stapledon 2009) Many more inequalities for the hj’s arising fromKneser’s Theorem (arXiv:0904.3035)
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 7
![Page 25: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/25.jpg)
General Properties of Ehrhart h-Vectors
A triangulation τ of P is unimodular if for any simplex of τ with verticesv0, v1, . . . , vd, the vectors v1 − v0, . . . , vd − v0 form a basis of Zd.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 8
![Page 26: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/26.jpg)
General Properties of Ehrhart h-Vectors
A triangulation τ of P is unimodular if for any simplex of τ with verticesv0, v1, . . . , vd, the vectors v1 − v0, . . . , vd − v0 form a basis of Zd.
The h-polynomial (h-vector) of a triangulation τ encodes the faces numbersof the simplices in τ of different dimensions.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 8
![Page 27: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/27.jpg)
General Properties of Ehrhart h-Vectors
A triangulation τ of P is unimodular if for any simplex of τ with verticesv0, v1, . . . , vd, the vectors v1 − v0, . . . , vd − v0 form a basis of Zd.
The h-polynomial (h-vector) of a triangulation τ encodes the faces numbersof the simplices in τ of different dimensions.
I (Stanley 1980) If P admits a unimodular triangulation then h(t) =(1− t)d+1 EhrP(t) is the h-polynomial of the triangulation.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 8
![Page 28: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/28.jpg)
General Properties of Ehrhart h-Vectors
A triangulation τ of P is unimodular if for any simplex of τ with verticesv0, v1, . . . , vd, the vectors v1 − v0, . . . , vd − v0 form a basis of Zd.
The h-polynomial (h-vector) of a triangulation τ encodes the faces numbersof the simplices in τ of different dimensions.
I (Stanley 1980) If P admits a unimodular triangulation then h(t) =(1− t)d+1 EhrP(t) is the h-polynomial of the triangulation.
I Recent papers of Reiner–Welker and Athanasiadis use this as a startingpoint to give conditions under which the Ehrhart h-vector is unimodal,i.e., hd ≤ hd−1 ≤ · · · ≤ hk ≥ hk−1 ≥ · · · ≥ h0 for some k.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 8
![Page 29: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/29.jpg)
The Main Question
Define h0(n), h1(n), . . . , hd(n) through
EhrnP(t) =hd(n) td + hd−1(n) td−1 + · · ·+ h0(n)
(1− t)d+1.
What does the Ehrhart h-vector (h0(n), h1(n), . . . , hd(n)) look like as nincreases?
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 9
![Page 30: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/30.jpg)
The Main Question
Define h0(n), h1(n), . . . , hd(n) through
EhrnP(t) =hd(n) td + hd−1(n) td−1 + · · ·+ h0(n)
(1− t)d+1.
What does the Ehrhart h-vector (h0(n), h1(n), . . . , hd(n)) look like as nincreases?
Let h(t) = (1− t)d+1 EhrP(t). The operator Un defined through
EhrnP(t) = 1 +∑m≥1
LP(nm) tm =Un h(t)
(1− t)d+1
appears in Number Theory as a Hecke operator and in Commutative Algebrain Veronese subring constructions.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 9
![Page 31: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/31.jpg)
Motivation I: Unimodular Triangulations
Theorem (Kempf–Knudsen–Mumford–Saint-Donat–Waterman 1970’s)For every lattice polytope P there exists an integer m such that mP admitsa regular unimodular triangulation.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 10
![Page 32: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/32.jpg)
Motivation I: Unimodular Triangulations
Theorem (Kempf–Knudsen–Mumford–Saint-Donat–Waterman 1970’s)For every lattice polytope P there exists an integer m such that mP admitsa regular unimodular triangulation.
Conjectures
(a) For every lattice polytope P there exists an integer m such that kPadmits a regular unimodular triangulation for k ≥ m.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 10
![Page 33: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/33.jpg)
Motivation I: Unimodular Triangulations
Theorem (Kempf–Knudsen–Mumford–Saint-Donat–Waterman 1970’s)For every lattice polytope P there exists an integer m such that mP admitsa regular unimodular triangulation.
Conjectures
(a) For every lattice polytope P there exists an integer m such that kPadmits a regular unimodular triangulation for k ≥ m.
(b) For every d there exists an integer md such that, if P is a d-dimensionallattice polytope, then mdP admits a regular unimodular triangulation.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 10
![Page 34: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/34.jpg)
Motivation I: Unimodular Triangulations
Theorem (Kempf–Knudsen–Mumford–Saint-Donat–Waterman 1970’s)For every lattice polytope P there exists an integer m such that mP admitsa regular unimodular triangulation.
Conjectures
(a) For every lattice polytope P there exists an integer m such that kPadmits a regular unimodular triangulation for k ≥ m.
(b) For every d there exists an integer md such that, if P is a d-dimensionallattice polytope, then mdP admits a regular unimodular triangulation.
(c) For every d there exists an integer md such that, if P is a d-dimensionallattice polytope, then kP admits a regular unimodular triangulation fork ≥ md.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 10
![Page 35: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/35.jpg)
Motivation II: Unimodal Ehrhart h-Vectors
Theorem (Athanasiadis–Hibi–Stanley 2004) If the d -dimensional latticepolytope P admits a regular unimodular triangulation, then the Ehrharth-vector of P satisfies
(a) hj+1 ≥ hd−j for 0 ≤ j ≤ bd2c − 1 ,
(b) hbd+12 c≥ hbd+1
2 c+1≥ · · · ≥ hd−1 ≥ hd ,
(c) hj ≤(h1+j−1
j
)for 0 ≤ j ≤ d .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 11
![Page 36: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/36.jpg)
Motivation II: Unimodal Ehrhart h-Vectors
Theorem (Athanasiadis–Hibi–Stanley 2004) If the d -dimensional latticepolytope P admits a regular unimodular triangulation, then the Ehrharth-vector of P satisfies
(a) hj+1 ≥ hd−j for 0 ≤ j ≤ bd2c − 1 ,
(b) hbd+12 c≥ hbd+1
2 c+1≥ · · · ≥ hd−1 ≥ hd ,
(c) hj ≤(h1+j−1
j
)for 0 ≤ j ≤ d .
In particular, if the Ehrhart h-vector of P is symmetric and P admits aregular unimodular triangulation, then the Ehrhart h-vector is unimodal.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 11
![Page 37: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/37.jpg)
Motivation II: Unimodal Ehrhart h-Vectors
Theorem (Athanasiadis–Hibi–Stanley 2004) If the d -dimensional latticepolytope P admits a regular unimodular triangulation, then the Ehrharth-vector of P satisfies
(a) hj+1 ≥ hd−j for 0 ≤ j ≤ bd2c − 1 ,
(b) hbd+12 c≥ hbd+1
2 c+1≥ · · · ≥ hd−1 ≥ hd ,
(c) hj ≤(h1+j−1
j
)for 0 ≤ j ≤ d .
In particular, if the Ehrhart h-vector of P is symmetric and P admits aregular unimodular triangulation, then the Ehrhart h-vector is unimodal.
There are (many) lattice polytopes for which (some of these) inequalitiesfail and one may hope to use this theorem to construct a counter-exampleto the Knudsen–Mumford–Waterman Conjectures.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 11
![Page 38: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/38.jpg)
Veronese Polynomials Are Eventually Unimodal
Theorem (Brenti–Welker 2008) For any d ∈ Z>0, there exists real numbersα1 < α2 < · · · < αd−1 < αd = 0 , such that, if h(t) is a polynomialof degree at most d with nonnegative integer coefficients and constantterm 1 , then for n sufficiently large, Un h(t) has negative real rootsβ1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 and lim
n→∞βj(n) = αj.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 12
![Page 39: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/39.jpg)
Veronese Polynomials Are Eventually Unimodal
Theorem (Brenti–Welker 2008) For any d ∈ Z>0, there exists real numbersα1 < α2 < · · · < αd−1 < αd = 0 , such that, if h(t) is a polynomialof degree at most d with nonnegative integer coefficients and constantterm 1 , then for n sufficiently large, Un h(t) has negative real rootsβ1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 and lim
n→∞βj(n) = αj.
Here “sufficiently large” depends on h(t).
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 12
![Page 40: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/40.jpg)
Veronese Polynomials Are Eventually Unimodal
Theorem (Brenti–Welker 2008) For any d ∈ Z>0, there exists real numbersα1 < α2 < · · · < αd−1 < αd = 0 , such that, if h(t) is a polynomialof degree at most d with nonnegative integer coefficients and constantterm 1 , then for n sufficiently large, Un h(t) has negative real rootsβ1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 and lim
n→∞βj(n) = αj.
Here “sufficiently large” depends on h(t).
If the polynomial p(t) = adtd +ad−1t
d−1 + · · ·+a0 has negative roots, thenits coefficients are (strictly) log concave (a2
j > aj−1aj+1).
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 12
![Page 41: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/41.jpg)
Veronese Polynomials Are Eventually Unimodal
Theorem (Brenti–Welker 2008) For any d ∈ Z>0, there exists real numbersα1 < α2 < · · · < αd−1 < αd = 0 , such that, if h(t) is a polynomialof degree at most d with nonnegative integer coefficients and constantterm 1 , then for n sufficiently large, Un h(t) has negative real rootsβ1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 and lim
n→∞βj(n) = αj.
Here “sufficiently large” depends on h(t).
If the polynomial p(t) = adtd +ad−1t
d−1 + · · ·+a0 has negative roots, thenits coefficients are (strictly) log concave (a2
j > aj−1aj+1) which, in turn,implies that the coefficients are (strictly) unimodal (ad < ad−1 < · · · <ak > ak−1 > · · · > a0 for some k).
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 12
![Page 42: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/42.jpg)
A General Theorem
The Eulerian polynomial Ad(t) is defined through∑m≥0
md tm =Ad(t)
(1− t)d+1.
Theorem (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of Ad(t) . There exist M,N depending only ond such that, if h(t) is a polynomial of degree at most d with nonnegativeinteger coefficients and constant term 1 , then for n ≥ N , Un h(t) hasnegative real roots β1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 withlim
n→∞βj(n) = ρj, and the coefficients of Un h(t) satisfy hj(n) < M hd(n).
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 13
![Page 43: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/43.jpg)
A General Theorem
The Eulerian polynomial Ad(t) is defined through∑m≥0
md tm =Ad(t)
(1− t)d+1.
Theorem (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of Ad(t) . There exist M,N depending only ond such that, if h(t) is a polynomial of degree at most d with nonnegativeinteger coefficients and constant term 1 , then for n ≥ N , Un h(t) hasnegative real roots β1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 withlim
n→∞βj(n) = ρj, and the coefficients of Un h(t) satisfy hj(n) < M hd(n).
In particular, the coefficients of Un h(t) are unimodal for n ≥ N .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 13
![Page 44: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/44.jpg)
A General Theorem
The Eulerian polynomial Ad(t) is defined through∑m≥0
md tm =Ad(t)
(1− t)d+1.
Theorem (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of Ad(t) . There exist M,N depending only ond such that, if h(t) is a polynomial of degree at most d with nonnegativeinteger coefficients and constant term 1 , then for n ≥ N , Un h(t) hasnegative real roots β1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 withlim
n→∞βj(n) = ρj, and the coefficients of Un h(t) satisfy hj(n) < M hd(n).
In particular, the coefficients of Un h(t) are unimodal for n ≥ N .
Furthermore, if h0 + · · ·+hj+1 ≥ hd + · · ·+hd−j for 0 ≤ j ≤⌊
d2
⌋− 1, with
at least one strict inequality, then we may choose N such that, for n ≥ N ,
h0 = h0(n) < hd(n) < h1(n) < · · · < hj(n) < hd−j(n) < hj+1(n)
< · · · < hbd+12 c
(n) < M hd(n) .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 13
![Page 45: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/45.jpg)
An Ehrhartian Corollary
Corollary (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of the Eulerian polynomial Ad(t). There exist M,Ndepending only on d such that, if P is a d-dimensional lattice polytope withEhrhart series numerator h(t), then for n ≥ N , Un h(t) has negative realroots β1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 with lim
n→∞βj(n) = ρj.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 14
![Page 46: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/46.jpg)
An Ehrhartian Corollary
Corollary (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of the Eulerian polynomial Ad(t). There exist M,Ndepending only on d such that, if P is a d-dimensional lattice polytope withEhrhart series numerator h(t), then for n ≥ N , Un h(t) has negative realroots β1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 with lim
n→∞βj(n) = ρj.
In particular, the coefficients of Un h(t) are unimodal for n ≥ N .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 14
![Page 47: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/47.jpg)
An Ehrhartian Corollary
Corollary (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of the Eulerian polynomial Ad(t). There exist M,Ndepending only on d such that, if P is a d-dimensional lattice polytope withEhrhart series numerator h(t), then for n ≥ N , Un h(t) has negative realroots β1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 with lim
n→∞βj(n) = ρj.
In particular, the coefficients of Un h(t) are unimodal for n ≥ N .
Furthermore, they satisfy
1 = h0(n) < hd(n) < h1(n) < · · · < hj(n) < hd−j(n) < hj+1(n)
< · · · < hbd+12 c
(n) < M hd(n) .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 14
![Page 48: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/48.jpg)
Ingredients I
Stapledon’s Decomposition A polynomial h(t) = hd+1td+1 +hdt
d + · · ·+h0
of degree at most d+1 has a unique decomposition h(t) = a(t)+b(t) wherea(t) and b(t) are polynomials satisfying a(t) = td a(1
t) and b(t) = td+1 b(1t).
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 15
![Page 49: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/49.jpg)
Ingredients I
Stapledon’s Decomposition A polynomial h(t) = hd+1td+1 +hdt
d + · · ·+h0
of degree at most d+1 has a unique decomposition h(t) = a(t)+b(t) wherea(t) and b(t) are polynomials satisfying a(t) = td a(1
t) and b(t) = td+1 b(1t).
I The coefficients of a(t) are positive if and only if h0 + · · · + hj ≥hd+1 + · · ·+ hd+1−j for 0 ≤ j <
⌊d2
⌋.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 15
![Page 50: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/50.jpg)
Ingredients I
Stapledon’s Decomposition A polynomial h(t) = hd+1td+1 +hdt
d + · · ·+h0
of degree at most d+1 has a unique decomposition h(t) = a(t)+b(t) wherea(t) and b(t) are polynomials satisfying a(t) = td a(1
t) and b(t) = td+1 b(1t).
I The coefficients of a(t) are positive if and only if h0 + · · · + hj ≥hd+1 + · · ·+ hd+1−j for 0 ≤ j <
⌊d2
⌋.
I The coefficients of a(t) are strictly unimodal if and only if hj+1 > hd−j
for 0 ≤ j ≤ bd2c − 1.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 15
![Page 51: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/51.jpg)
Ingredients I
Stapledon’s Decomposition A polynomial h(t) = hd+1td+1 +hdt
d + · · ·+h0
of degree at most d+1 has a unique decomposition h(t) = a(t)+b(t) wherea(t) and b(t) are polynomials satisfying a(t) = td a(1
t) and b(t) = td+1 b(1t).
I The coefficients of a(t) are positive if and only if h0 + · · · + hj ≥hd+1 + · · ·+ hd+1−j for 0 ≤ j <
⌊d2
⌋.
I The coefficients of a(t) are strictly unimodal if and only if hj+1 > hd−j
for 0 ≤ j ≤ bd2c − 1.
Theorem (Stapledon 2008) If h(t) is the Ehrhart h-vector of a latticed -polytope, then the coefficients of a(t) satisfy 1 = a0 ≤ a1 ≤ aj for2 ≤ j ≤ d− 1.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 15
![Page 52: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/52.jpg)
Ingredients I
Stapledon’s Decomposition A polynomial h(t) = hd+1td+1 +hdt
d + · · ·+h0
of degree at most d+1 has a unique decomposition h(t) = a(t)+b(t) wherea(t) and b(t) are polynomials satisfying a(t) = td a(1
t) and b(t) = td+1 b(1t).
I The coefficients of a(t) are positive if and only if h0 + · · · + hj ≥hd+1 + · · ·+ hd+1−j for 0 ≤ j <
⌊d2
⌋.
I The coefficients of a(t) are strictly unimodal if and only if hj+1 > hd−j
for 0 ≤ j ≤ bd2c − 1.
Theorem (Stapledon 2008) If h(t) is the Ehrhart h-vector of a latticed -polytope, then the coefficients of a(t) satisfy 1 = a0 ≤ a1 ≤ aj for2 ≤ j ≤ d− 1.
Corollary Hibi’s inequalities h0+ · · ·+hj+1 ≥ hd+ · · ·+hd−j for the Ehrharth-vector are strict.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 15
![Page 53: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/53.jpg)
Ingredients II
Let h(t) = hd+1td+1 + hdt
d + · · · + h0 be a polynomial of degree at most
d + 1 and expand h(t)
(1−t)d+1 = h0 +∑
m≥1 g(m) tm, for some polynomial
g(m) = gdmd + gd−1m
d−1 + · · ·+ g0.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 16
![Page 54: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/54.jpg)
Ingredients II
Let h(t) = hd+1td+1 + hdt
d + · · · + h0 be a polynomial of degree at most
d + 1 and expand h(t)
(1−t)d+1 = h0 +∑
m≥1 g(m) tm, for some polynomial
g(m) = gdmd + gd−1m
d−1 + · · ·+ g0.
Theorem (Betke–McMullen 1985) If hj ≥ 0 for 0 ≤ j ≤ d+ 1, then for any1 ≤ r ≤ d− 1,
gr ≤ (−1)d−rSr(d) gd +(−1)d−r−1 h0 Sr+1(d)
(d− 1)!,
where Si(d) is the Stirling number of the first kind.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 16
![Page 55: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/55.jpg)
Ingredients II
Let h(t) = hd+1td+1 + hdt
d + · · · + h0 be a polynomial of degree at most
d + 1 and expand h(t)
(1−t)d+1 = h0 +∑
m≥1 g(m) tm, for some polynomial
g(m) = gdmd + gd−1m
d−1 + · · ·+ g0.
Theorem (Betke–McMullen 1985) If hj ≥ 0 for 0 ≤ j ≤ d+ 1, then for any1 ≤ r ≤ d− 1,
gr ≤ (−1)d−rSr(d) gd +(−1)d−r−1 h0 Sr+1(d)
(d− 1)!,
where Si(d) is the Stirling number of the first kind.
Theorem (MB–Stapledon) If h0 + · · · + hj ≥ hd+1 + · · · + hd+1−j for0 ≤ j ≤
⌊d2
⌋, with at least one strict inequality, then
gd−1−2r ≤ Sd−1−2r(d− 1) gd−1 −(h0 − hd+1)Sd−2r(d− 1)
2(d− 2)!.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 16
![Page 56: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/56.jpg)
Ingredients III
Let h(t) = hdmd +hd−1m
d−1 + · · ·+h0 be a polynomial of degree at most
d and expand h(t)
(1−t)d+1 =∑
m≥0 g(m) tm, for some polynomial g(m) =gdm
d + gd−1md−1 + · · ·+ g0.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 17
![Page 57: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/57.jpg)
Ingredients III
Let h(t) = hdmd +hd−1m
d−1 + · · ·+h0 be a polynomial of degree at most
d and expand h(t)
(1−t)d+1 =∑
m≥0 g(m) tm, for some polynomial g(m) =gdm
d + gd−1md−1 + · · ·+ g0.
Recall our notation Un h(t) = hd(n) td + hd−1(n) td−1 + · · ·+ h0(n).
Lemma Un h(t) =d∑
j=0
gj Aj(t) (1− t)d−j nj.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 17
![Page 58: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/58.jpg)
Ingredients III
Let h(t) = hdmd +hd−1m
d−1 + · · ·+h0 be a polynomial of degree at most
d and expand h(t)
(1−t)d+1 =∑
m≥0 g(m) tm, for some polynomial g(m) =gdm
d + gd−1md−1 + · · ·+ g0.
Recall our notation Un h(t) = hd(n) td + hd−1(n) td−1 + · · ·+ h0(n).
Lemma Un h(t) =d∑
j=0
gj Aj(t) (1− t)d−j nj.
In particular, for 1 ≤ j ≤ d, hj(n) is a polynomial in n of degree d and
hj(n) = A(d, j) gd nd+(A(d−1, j)−A(d−1, j−1)) gd−1 n
d−1+O(nd−2) .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 17
![Page 59: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/59.jpg)
Ingredients IV
Lemma Un h(t) =d∑
j=0
gj Aj(t) (1− t)d−j nj.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 18
![Page 60: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/60.jpg)
Ingredients IV
Lemma Un h(t) =d∑
j=0
gj Aj(t) (1− t)d−j nj.
Exercise The nonzero roots of the Eulerian polynomial Ad(t) =∑dj=0A(d, j) tj are negative.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 18
![Page 61: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/61.jpg)
Ingredients IV
Lemma Un h(t) =d∑
j=0
gj Aj(t) (1− t)d−j nj.
Exercise The nonzero roots of the Eulerian polynomial Ad(t) =∑dj=0A(d, j) tj are negative.
Theorem (Cauchy) Let p(n) = pd nd +pd−1 n
d−1 + · · ·+p0 be a polynomialof degree d with real coefficients. The complex roots of p(n) lie in the opendisc {
z ∈ C : |z| < 1 + max0≤j≤d
∣∣∣∣pj
pd
∣∣∣∣} .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 18
![Page 62: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/62.jpg)
Veronese Subrings
Let R = ⊕j≥0Rj be a graded ring; we assume that R0 = K is a field andthat R is finitely generated over K.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 19
![Page 63: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/63.jpg)
Veronese Subrings
Let R = ⊕j≥0Rj be a graded ring; we assume that R0 = K is a field andthat R is finitely generated over K.
R〈n〉 :=⊕j≥0
Rjn (nth Veronese subring of R)
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 19
![Page 64: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/64.jpg)
Veronese Subrings
Let R = ⊕j≥0Rj be a graded ring; we assume that R0 = K is a field andthat R is finitely generated over K.
R〈n〉 :=⊕j≥0
Rjn (nth Veronese subring of R)
H(R,m) := dimK Rm (Hilbert function of R)
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 19
![Page 65: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/65.jpg)
Veronese Subrings
Let R = ⊕j≥0Rj be a graded ring; we assume that R0 = K is a field andthat R is finitely generated over K.
R〈n〉 :=⊕j≥0
Rjn (nth Veronese subring of R)
H(R,m) := dimK Rm (Hilbert function of R)
By a theorem of Hilbert H(R,m) is a polynomial in m when m is sufficientlylarge. Note that
Un h(t)(1− t)d+1
=∑m≥0
H(R〈n〉,m) tm.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 19
![Page 66: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/66.jpg)
Veronese Subrings
Let R = ⊕j≥0Rj be a graded ring; we assume that R0 = K is a field andthat R is finitely generated over K.
R〈n〉 :=⊕j≥0
Rjn (nth Veronese subring of R)
H(R,m) := dimK Rm (Hilbert function of R)
By a theorem of Hilbert H(R,m) is a polynomial in m when m is sufficientlylarge. Note that
Un h(t)(1− t)d+1
=∑m≥0
H(R〈n〉,m) tm.
Example Denote the cone over P × {1} by coneP . Then the semigroupalgebra K[coneP ∩Zd+1] (graded by the projection to the last coordinate)gives rise to the Hilbert function H(K[coneP ∩ Zd+1],m) = LP(m).
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 19
![Page 67: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/67.jpg)
A Veronese Corollary
Corollary (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of the Eulerian polynomial Ad(t). There exist M,N
depending only on d such that, if R =⊕j≥0
Rj is a finitely generated graded
ring over a field R0 = K that is Cohen–Macauley and module finite overthe K -subalgebra generated by R1 , and if the Hilbert function H(R,m)is a polynomial in m , then for n ≥ N , Un h(t) has negative real rootsβ1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 with lim
n→∞βj(n) = ρj.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 20
![Page 68: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/68.jpg)
A Veronese Corollary
Corollary (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of the Eulerian polynomial Ad(t). There exist M,N
depending only on d such that, if R =⊕j≥0
Rj is a finitely generated graded
ring over a field R0 = K that is Cohen–Macauley and module finite overthe K -subalgebra generated by R1 , and if the Hilbert function H(R,m)is a polynomial in m , then for n ≥ N , Un h(t) has negative real rootsβ1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 with lim
n→∞βj(n) = ρj.
In particular, the coefficients of Un h(t) are unimodal for n ≥ N .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 20
![Page 69: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/69.jpg)
A Veronese Corollary
Corollary (MB–Stapledon) Fix a positive integer d and let ρ1 < ρ2 < · · · <ρd = 0 denote the roots of the Eulerian polynomial Ad(t). There exist M,N
depending only on d such that, if R =⊕j≥0
Rj is a finitely generated graded
ring over a field R0 = K that is Cohen–Macauley and module finite overthe K -subalgebra generated by R1 , and if the Hilbert function H(R,m)is a polynomial in m , then for n ≥ N , Un h(t) has negative real rootsβ1(n) < β2(n) < · · · < βd−1(n) < βd(n) < 0 with lim
n→∞βj(n) = ρj.
In particular, the coefficients of Un h(t) are unimodal for n ≥ N .
Furthermore, they satisfy hj(n) < M hd(n) for 0 ≤ j ≤ n and n ≥ N .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 20
![Page 70: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/70.jpg)
Open Problems
Find optimal choices for M and N in any of our theorems.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 21
![Page 71: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/71.jpg)
Open Problems
Find optimal choices for M and N in any of our theorems.
Conjecture For Ehrhart series of d-dimensional polytopes, N = d.
(Open for d ≥ 3)
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 21
![Page 72: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/72.jpg)
One Result about Explicit Bounds
Recall our inequalities hj+1(n) > hd−j(n) in the main theorem. . .
Theorem (MB–Stapledon) Fix a positive integer d and set N = d if d iseven and N = d+1
2 if d is odd. If h(t) is a polynomial of degree at most d
satisfying h0 + · · ·+ hj+1 > hd + · · ·+ hd−j for 0 ≤ j ≤⌊
d2
⌋− 1, then the
coefficients of Un h(t) satisfy hj+1(n) > hd−j(n) for 0 ≤ j ≤⌊
d2
⌋− 1 and
n ≥ N .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 22
![Page 73: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/73.jpg)
One Result about Explicit Bounds
Recall our inequalities hj+1(n) > hd−j(n) in the main theorem. . .
Theorem (MB–Stapledon) Fix a positive integer d and set N = d if d iseven and N = d+1
2 if d is odd. If h(t) is a polynomial of degree at most d
satisfying h0 + · · ·+ hj+1 > hd + · · ·+ hd−j for 0 ≤ j ≤⌊
d2
⌋− 1, then the
coefficients of Un h(t) satisfy hj+1(n) > hd−j(n) for 0 ≤ j ≤⌊
d2
⌋− 1 and
n ≥ N .
Corollary Fix a positive integer d and set N = d if d is even and N = d+12
if d is odd. If P is a d -dimensional lattice polytope with Ehrhart h-vector h(t), then the coefficients of Un h(t) satisfy hj+1(n) > hd−j(n) for0 ≤ j ≤
⌊d2
⌋− 1 and n ≥ N .
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 22
![Page 74: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/74.jpg)
The Message
The Ehrhart series of nP becomes friendlier as n increases.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 23
![Page 75: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/75.jpg)
The Message
The Ehrhart series of nP becomes friendlier as n increases.
In fixed dimension, you don’t have to wait forever to make all Ehrhart serieslook friendly.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 23
![Page 76: Asymptotics of Ehrhart Series of Lattice Polytopesmath.sfsu.edu/beck/papers/asymptotics.slides.pdf · 2009-09-16 · system measures some fundamental data of this system (\average").](https://reader034.fdocuments.in/reader034/viewer/2022050421/5f90bb606db2b762506ef8c4/html5/thumbnails/76.jpg)
The Message
The Ehrhart series of nP becomes friendlier as n increases.
In fixed dimension, you don’t have to wait forever to make all Ehrhart serieslook friendly.
Homework Figure out what all of this has to do with carrying digits whensumming 100-digit numbers.
Asymptotics of Ehrhart Series of Lattice Polytopes Matthias Beck 23