Polytopes, Ehrhart theory and simplicial complexes

Face numbers and h-vectors of simplicial spheres and polytopes (Dehn–Sommerville, upper bound theorem), Cohen–Macaulay, partitionable and interval-decomposable complexes, simplicial homology, Ehrhart polynomials, h*-vectors and the integer decomposition property, lattice parallelograms, and hyperplane arrangements.

DefinitionOrdinary Ehrhart count#✓fully proved

For a natural number \(d\), a set \(P \subseteq \mathbb{Q}^d\) and a natural number \(m\), the ordinary Ehrhart count is the number \(|mP \cap \mathbb{Z}^d|\) of integer points in the dilate \(mP = \{m x : x \in P\}\), where \(\mathbb{Z}^d\) is viewed inside \(\mathbb{Q}^d\); it is \(0\) when \(mP \cap \mathbb{Z}^d\) is infinite.

For polytopes the intersection is finite.

noncomputable def ordinaryEhrhartCount (d : ℕ) (P : Set (Fin d → ℚ)) (m : ℕ) : ℕ :=
  Set.ncard
    (((fun x : Fin d → ℚ => (m : ℚ) • x) '' P) ∩
      Set.range fun z : Fin d → ℤ => fun i => (z i : ℚ))

Sources

• Ryuichi Sakamoto, "The h*-Polynomial of the Cut Polytope of K_{2,m} in the Lattice Spanned by its Vertices," Journal of Integer Sequences 23 (2020), Article 20.7.5.

Used by 1

Used byPolynomial counting the lattice points of dilates

• Clean lattice parallelograms
• Face numbers of simplicial spheres
• Colour groups of sphere triangulations
• Rational homology of simplicial complexes
• Partitionable and interval-decomposable complexes
• Upper bound theorem for simplicial spheres
• Weighted projective space simplices
• Ehrhart polynomials with roots on the canonical line
• Unimodality of h*-vectors

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