Polytopes With Large Signature Joint work with Michael Joswig - PowerPoint PPT Presentation

Polytopes With Large Signature Joint work with Michael Joswig Nikolaus Witte TU-Berlin / TU-Darmstadt witte@math.tu-berlin.de Algebraic and Geometric Combinatorics, Anogia 2005

Outline Introduction 1 Motivation The Staircase Triangulation Triangulating Products Of Polytopes 2 The Simplicial Product The Product Theorem Signature of the d -Cube 3 Lower Bounds Upper Bounds Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 2 / 30

Outline Introduction 1 Motivation The Staircase Triangulation Triangulating Products Of Polytopes 2 The Simplicial Product The Product Theorem Signature of the d -Cube 3 Lower Bounds Upper Bounds Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 2 / 30

Outline Introduction 1 Motivation The Staircase Triangulation Triangulating Products Of Polytopes 2 The Simplicial Product The Product Theorem Signature of the d -Cube 3 Lower Bounds Upper Bounds Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 2 / 30

Outline Introduction 1 Motivation The Staircase Triangulation Triangulating Products Of Polytopes 2 The Simplicial Product The Product Theorem Signature of the d -Cube 3 Lower Bounds Upper Bounds Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 3 / 30

Real Solutions of Polynomial Systems A generic system S F 1 ( t 1 , . . . , t n ) = . . . = F n ( t 1 , . . . , t n ) = 0 of n real polynomial equations has finitely many real solutions. In general it is extremely difficult to compute the real solutions of S . Not even the number of real solutions can be computed easily, or in fact if there are any solutions at all. F (x,y) 2 F = F 2 F = F 1 1 2 z=0 F (x,y) 1 Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 4 / 30

Real Solutions of Polynomial Systems Theorem (S OPRUNOVA & S OTTILE ’04) Let N be a lattice polytope and let N ω be a convex and balanced triangulation of N . Then there is an associated system S ( N ω ) of real polynomial equations and the number of real solutions of S ( N ω ) is at least the signature σ ( N ω ) of N ω . Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 5 / 30

Real Solutions of Polynomial Systems Theorem (S OPRUNOVA & S OTTILE ’04) Let N be a lattice polytope and let N ω be a convex and balanced triangulation of N . Then there is an associated system S ( N ω ) of real polynomial equations and the number of real solutions of S ( N ω ) is at least the signature σ ( N ω ) of N ω . Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 6 / 30

Real Solutions of Polynomial Systems Theorem (S OPRUNOVA & S OTTILE ’04) Let N be a lattice polytope and let N ω be a convex and balanced triangulation of N . Then there is an associated system S ( N ω ) of real polynomial equations and the number of real solutions of S ( N ω ) is at least the signature σ ( N ω ) of N ω . Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 7 / 30

Real Solutions of Polynomial Systems Theorem (S OPRUNOVA & S OTTILE ’04) Let N be a lattice polytope and let N ω be a convex and balanced triangulation of N . Then there is an associated system S ( N ω ) of real polynomial equations and the number of real solutions of S ( N ω ) is at least the signature σ ( N ω ) of N ω . Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 8 / 30

Real Solutions of Polynomial Systems Theorem (S OPRUNOVA & S OTTILE ’04) Let N be a lattice polytope and let N ω be a convex and balanced triangulation of N . Then there is an associated system S ( N ω ) of real polynomial equations and the number of real solutions of S ( N ω ) is at least the signature σ ( N ω ) of N ω . Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 9 / 30

Polytopes With Large Signature It is extremely difficult to determine the number of real solutions of a polynomial system. S OPRUNOVA & S OTTILE construct non trivial polynomial systems where the number of real solutions is bounded from below by the signature of a triangulation of the Newton Polytope. We want to construct lattice polytopes with large signature. Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 10 / 30

Polytopes With Large Signature It is extremely difficult to determine the number of real solutions of a polynomial system. S OPRUNOVA & S OTTILE construct non trivial polynomial systems where the number of real solutions is bounded from below by the signature of a triangulation of the Newton Polytope. We want to construct lattice polytopes with large signature. Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 10 / 30

Polytopes With Large Signature It is extremely difficult to determine the number of real solutions of a polynomial system. S OPRUNOVA & S OTTILE construct non trivial polynomial systems where the number of real solutions is bounded from below by the signature of a triangulation of the Newton Polytope. We want to construct lattice polytopes with large signature. Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 10 / 30

The Staircase Triangulation The facet ( 0 , 1 , 0 , 0 , 1 ) of stc (∆ 2 × ∆ 3 ) . ∆ 3 The triangulation stc (∆ 2 × ∆ 3 ) � 2 + 3 � has = 10 facets. 2 ∆ 2 Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 11 / 30

The Staircase Triangulation The staircase triangulation is a lattice triangulation, ∆ 3 convex, and balanced. ∆ 2 Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 12 / 30

Example: stc (∆ 1 × ∆ 2 ) ∆ 2 ∆ 1 Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 13 / 30

Signature Theorem (S TANLEY ’97, S OPRUNOVA & S OTTILE ’04) The signature of the staircase triangulation is � k + l � = σ 2 k , 2 l k � k + l � = σ 2 k , 2 l + 1 k = 0 σ 2 k + 1 , 2 l + 1 Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 14 / 30

Outline Introduction 1 Motivation The Staircase Triangulation Triangulating Products Of Polytopes 2 The Simplicial Product The Product Theorem Signature of the d -Cube 3 Lower Bounds Upper Bounds Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 15 / 30

The Simplicial Product Idea: Triangulating the product K × L of two abstract simplicial complexes by using staircase triangulations for the cells of K × L . Problem: How do the triangulated facets fit together? Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 16 / 30

Definition A facet of the simplicial product K × stc L . Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 17 / 30

The Intersection of 2 Facets The intersection of two facets of K × stc L . Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 18 / 30

The Vertex Ordering Does Matter Different vertex orderings may yield different triangulations of K × L . Given a “wrong” ordering, the simplicial product of two balanced complexes might not be balanced. Lemma If K and L are balanced simplicial complexes with color consecutive vertex orderings then K × stc L is again balanced. Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 19 / 30

The Vertex Ordering Does Matter Different vertex orderings may yield different triangulations of K × L . Given a “wrong” ordering, the simplicial product of two balanced complexes might not be balanced. Lemma If K and L are balanced simplicial complexes with color consecutive vertex orderings then K × stc L is again balanced. Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 19 / 30

Example: 3 Triangulations of the 3-Cube 0 2 0 3 1 3 1 3 2 1 0 2 Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 20 / 30

Regularity Lemma If K and L are regular simplicial complexes then K × stc L is regular for any vertex orderings of K and L. Let λ : R m → R and µ : R n → R be lifting functions of K resp. L . Define a lifting function ω : R m + n → R by ω : R m + n → R ( v , w ) �→ λ ( v ) + µ ( w ) + ǫ ( v , w ) Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 21 / 30

Regularity Lemma If K and L are regular simplicial complexes then K × stc L is regular for any vertex orderings of K and L. Let λ : R m → R and µ : R n → R be lifting functions of K resp. L . Define a lifting function ω : R m + n → R by ω : R m + n → R ( v , w ) �→ λ ( v ) + µ ( w ) + ǫ ( v , w ) Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 21 / 30

The Product Theorem Theorem (J OSWIG & W ’05) Let K and L be convex and balanced simplicial complexes of dimension m resp. n. Then K × stc L is a convex and balanced triangulation for any color consecutive vertex orderings of K and L. The signature of K × stc L is σ ( K × stc L ) = σ ( K ) σ ( L ) σ m , n . Corollary Let P and Q be lattice polytopes of dimension m resp. n. Let the signatures of P and Q be non-negative. Then the signature of P × Q is at least σ ( P × Q ) ≥ σ ( P ) σ ( Q ) σ m , n . Nikolaus Witte (TU-Berlin) Polytopes With Large Signature AGC 2005 22 / 30