Ehrhart theory for spanning lattice polytopes
Benjamin Nill
Otto-von-Guericke-Universität Magdeburg
x{x}T ,ξξπ(x)Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Overview
Motivation
Theorem and Applications
Proof for simplices
Half-open triangulations
Sketch of proof

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Notation
P ⊂ Rd d-dimensional lattice polytope∞Xdk|(kP) ∩ Z |t =k=0Pd∗ i
i=0 hi td+1(1 − t)hP∗ = (h0∗ , . . . , hs∗ , 0 . . . , 0) with hs∗ 6= 0,s = deg(P)Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Motivation: low degree
s=0

⇐⇒

P∼
= ∆d unimodular simplex

s = 1 : classification known (Batyrev, N. ’07)

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Motivation: vanishing of h∗ -coefficients
Question:hi∗ = 0

Benjamin Nill (OvGU Magdeburg)=⇒⇐?

Ehrhart theory for spanning lattice polytopes

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Motivation: i = 1h1∗ = 0⇐⇒

Benjamin Nill (OvGU Magdeburg)|P ∩ Zd+1 | = d + 1⇐⇒

Ehrhart theory for spanning lattice polytopes

P empty simplex

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Motivation: i = 1h1∗ = 0⇐⇒|P ∩ Zd+1 | = d + 1⇐⇒

P empty simplex

Let us give alternative formulation.

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Motivation: i = 1h1∗ = 0⇐⇒|P ∩ Zd+1 | = d + 1⇐⇒

P empty simplex

Let us give alternative formulation.

Definition
ΓP sublattice of Zd+1 generated by (P × {1}) ∩ Zd+1 .
P̃ × {1} is lattice polytope with respect to ΓP having the vertices of P .
In order to write P̃ as lattice polytope of Zd one has to pick affine latticebasis of the affine sublattice of Zd of integer affine combinations ofP ∩ Zd .

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Motivation: i = 1
P empty simplex

Benjamin Nill (OvGU Magdeburg)⇐⇒P̃ unimodular simplex

Ehrhart theory for spanning lattice polytopes

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Motivation: i = 1
P empty simplex⇐⇒P̃ unimodular simplex⇐⇒

deg(P̃) = 0

Alternative criterion:h1∗ = 0

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Motivation: i = 1
P empty simplex⇐⇒P̃ unimodular simplex⇐⇒deg(P̃) = 0

Alternative criterion:h1∗ = 0
Example:PP̃

(1, 1, 2)(0, 0, 1)(0, 1, 0)(1, 0, 0)

Benjamin Nill (OvGU Magdeburg)→(1, 0, 0)(0, 1, 0)Ehrhart theory for spanning lattice polytopes

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Motivation: i = 2
h1∗ = 0⇐⇒deg(P̃) = 0Proposition (Blekherman, Smith, Velasco ’13)h2∗ = 0 ⇐⇒ deg(P̃) ≤ 1 and P is 2-IDP
i-IDP: every lattice point in iP is a sum of i lattice points in P.
Is there a generalization to i ≥ 3?

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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The main theorem
P is spanning if and only if ΓP = Zd+1 .

Any lattice polytope is spanning after a change of lattice
P × {1} lattice polytope (with respect to Zd+1 ).
P̃ × {1} spanning lattice polytope (with respect to ΓP ) withsame lattice points (h̃1∗ = h1∗ )h̃i∗ ≤ hi∗
Example:P̃
(0, 0, 1)(1, 0, 0)(0, 1, 0)→(1, 0, 0)h̃∗ = (1, 0, 0, 0)
Benjamin Nill (OvGU Magdeburg)P(1, 1, 2)(0, 1, 0)h∗ = (1, 0, 1, 0)

Ehrhart theory for spanning lattice polytopes

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The main theorem
Example:P̃
(0, 0, 1)(1, 0, 0)

P(1, 1, 2)(0, 1, 0)→(1, 0, 0)h̃∗ = (1, 0, 0, 0)(0, 1, 0)h∗ = (1, 0, 1, 0)

Theorem A (Hofscheier, Katthän, N. ’16)
P spanning =⇒ h0∗ , . . . , hs∗ 6= 0 (no gaps)

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Back to our motivation: i ≥ 3
Proposition (Blekherman, Smith, Velasco ’13)h2∗ = 0 ⇐⇒ deg(P̃) ≤ 1 and P is 2-IDP

Proposition/Corollary to Theorem Adeg(P̃) ≤ i − 1 and P is i-IDP =⇒ hi∗ = 0 =⇒ deg(P̃) ≤ i − 1i≥3

Careful: hi∗ = 0 6=⇒ i-IDP.

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Back to our motivation: i ≥ 3
Proposition (Blekherman, Smith, Velasco ’13)h2∗ = 0 ⇐⇒ deg(P̃) ≤ 1 and P is 2-IDP

Proposition/Corollary to Theorem Adeg(P̃) ≤ i − 1 and P is i-IDP =⇒ hi∗ = 0 =⇒ deg(P̃) ≤ i − 1i≥3Careful: hi∗ = 0 6=⇒ i-IDP.
Geometric consequence:

Corollary to Theorem A and (Haase, N., Payne ’09)hi∗ = 0 and d ≥ 20(i − 1)2=⇒ d ≥ 20(deg(P̃))2=⇒ P̃ is a Cayley polytope=⇒ all lattice points of P lie on two parallel faces.
Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Relation to unimodality conjecture
Conjecture (Stanley ’89; Ohsugi, Hibi ’05; Schepers, Van Langenhoven ’11)
P IDP (integrally-closed)

Benjamin Nill (OvGU Magdeburg)=⇒h∗ is unimodal

Ehrhart theory for spanning lattice polytopes

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Relation to unimodality conjecture
Conjecture (Stanley ’89; Ohsugi, Hibi ’05; Schepers, Van Langenhoven ’11)
P IDP (integrally-closed) =⇒ h∗ is unimodal⇓⇓
P spanning=⇒ h∗ has no gapsd≥5

P spanning 6=⇒ h∗ is unimodal

Alternative algebraic proof when P
P IDP=⇒h∗ is M-sequence=⇒

no gaps(hi∗ = dimC Ri for R standard-graded C-algebra)

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Polyhedral consequence of Eisenbud-Goto
Eisenbud-Goto-Conjecturek alg. closed, I homogeneous prime ideal in S := k[x1 , . . . , xn ]. Thenreg(S/I ) + codim(S/I ) ≤ deg(S/I ).

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Eisenbud-Goto-Conjecture is wrong (?) (McCullough, Peeva ’16 (?))
k alg. closed, I homogeneous prime ideal in S := k[x1 , . . . , xn ]. Thenreg(S/I ) + codim(S/I ) ≤ deg(S/I ).

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Polyhedral consequence of Eisenbud-Goto
Eisenbud-Goto-Conjecture is wrong (?) (McCullough, Peeva ’16 (?))
k alg. closed, I homogeneous prime ideal in S := k[x1 , . . . , xn ]. Thenreg(S/I ) + codim(S/I ) ≤ deg(S/I ).
For P spanning (here, S/I is k-algebra generated by (P ×{1}) ∩ Zd+1 ):deg(S/I ) = d!vol(P) = h0∗ + h1∗ + h2∗ + · · · + hs∗
codim(S/I ) = h1∗
reg(S/I ) ≥ s

Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Polyhedral consequence of Eisenbud-Goto
Eisenbud-Goto-Conjecture is wrong (?) (McCullough, Peeva ’16 (?))
k alg. closed, I homogeneous prime ideal in S := k[x1 , . . . , xn ]. Thenreg(S/I ) + codim(S/I ) ≤ deg(S/I ).
For P spanning (here, S/I is k-algebra generated by (P ×{1}) ∩ Zd+1 ):deg(S/I ) = d!vol(P) = h0∗ + h1∗ + h2∗ + · · · + hs∗
codim(S/I ) = h1∗
reg(S/I ) ≥ s

Corollary to Theorem AP spanning=⇒s + h1∗ ≤ h0∗ + · · · + hs∗

Proof: r.h.s. ⇐⇒ s − 1 ≤ h2∗ + · · · + hs∗ ⇐⇒1 + · · · + 1 ≤ h2∗ + · · · + hs∗
Benjamin Nill (OvGU Magdeburg)Ehrhart theory for spanning lattice polytopes

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Polyhedral consequence of Eisenbud-Goto
Eisenbud-Goto-Conjecture is wrong (?) (McCullough, Peeva ’16 (?))
k alg. closed, I homogeneous prime ideal in S := k[x1 , . . . , xn ]. Thenreg(S/I ) + codim(S/I ) ≤ deg(S/I ).
For P spanning (here, S/I is k-algebra generated by (P ×{1}) ∩ Zd+1 ):deg(S/I ) = d!vol(P) = h0∗ + h1∗ + h2∗ + · · · + hs∗
codim(S/I ) = h1∗
reg(S/I ) ≥ s

Corollary to Theorem AP spanning=⇒s + h1∗ ≤ h0∗ + · · · + hs∗

Proof: r.h.s. ⇐⇒ s − 1 ≤ h2∗ + · · · + hs∗ ⇐⇒1 + · · · + 1 ≤ h2∗ + · · · + hs∗
Needs spanning: otherwise h∗ = (1, 0, . . . , 0, 1, 0, . . . , 0) possible!
Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Proof idea for spanning simplices
Any coset in Zd+1 modulo lattice spanned by vertices
has unique representative in Π ∩ Zd+1 :x ∈ Zd+1[x] ∈ Π ∩ Zd+1 .

Cx[x]

Benjamin Nill (OvGU Magdeburg)ΠEhrhart theory for spanning lattice polytopes

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Proof idea for spanning simplices
Let x ∈ Π ∩ Zd+1 with height(x) = s.
P spanning =⇒ exist lattice points in P × {1} such thatx = w1 + · · · − u1 − · · ·Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Proof idea for spanning simplices
Let x ∈ Π ∩ Zd+1 with height(x) = s.
P spanning =⇒ exist lattice points in P × {1} such thatx = w1 + · · · − u1 − · · ·Π ∩ Zd+1 is finite abelian group, so exists kj ∈ N≥1 such thatkj [uj ] = [0], so [−uj ] = (kj − 1)[uj ].
x = [x] = [w1 + · · · + wk ]Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Proof idea for spanning simplices
x = [x] = [w1 + · · · + wk ]Each step goes up at most by one - all intermediate values are hit.

Benjamin Nill (OvGU Magdeburg)

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Method of proof: half-open triangulations
The h∗ -vector counts lattice points in union of half-open parallelepipeds
x{x}T ,ξξπ(x)Benjamin Nill (OvGU Magdeburg)

Ehrhart theory for spanning lattice polytopes

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Key idea of proof

Benjamin Nill (OvGU Magdeburg)

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Using the circuit relation leads to step functions

Benjamin Nill (OvGU Magdeburg)

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Using the circuit relation leads to step functions

Benjamin Nill (OvGU Magdeburg)

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