Triangulations of point sets, high dimensional Polytopes and - PowerPoint PPT Presentation

Triangulations of point sets, high dimensional Polytopes and Applications Vissarion Fisikopoulos advisor: Professor Ioannis Emiris MPLA :: Graduate Program in Logic, Algorithms and Computation University of Athens 23 December 2009

Motivation

Outline Triangulations 1 Mixed Subdivisions 2 Resultant Polytopes 3

Outline Triangulations 1 Mixed Subdivisions 2 Resultant Polytopes 3

Polyhedral Subdivisions Definition A polyhedral subdivision of a point set ❆ in R ❞ is a collection ❙ of subsets of ❆ called cells s.t. • The cells cover ❝♦♥✈ ✭ ❆ ✮ • Every pair of cells intersect at a (possibly empty) common face A polyhedral subdivision of ❆ whose cells are all simplices, is called triangulation . A point set, two invalid subdivisions, a polyhedral subdivision and a triangulation.

Regular Polyhedral Subdivisions Definition A polyhedral subdivision P of a point set ❆ in R ❞ is regular if there exist a lifting function ✇ ✿ ❆ ✦ R s.t. P is the projection to R ❞ of the lower hull of ❡ ❆ ❂ ✭ ❛ ❀ ✇ ✭ ❛ ✮✮ ❀ ❛ ✷ ❆ . A point set with two regular triangulations and one with two non regular triangulations.

The Secondary Polytope Definition • Let ❚ be a triangulation of ❆ . Then the GKZ-vector of ❚ is ✟ ❆ ✭ ❚ ✮ ❂ P P ✛ ✷ ❚ ✿ ❛ ✐ ✷ ✛ ✈♦❧ ✭ ✛ ✮ ❡ ❛ ✐ ❛ ✐ ✷ ❆ • The Secondary polytope ✝✭ ❆ ✮ is the convex hull of GKZ-vectors for all triangulations of ❆ . vol = 1 vol = 2 vol = 2 A 1 2 3 4 T 1 Φ A ( T 1 ) = (5 , 0 , 0 , 5) Σ( A ) T 1 T 2 T 2 Φ A ( T 2 ) = (1 , 5 , 0 , 4) T 3 Φ A ( T 3 ) = (3 , 0 , 5 , 2) T 3 T 4 Φ A ( T 4 ) = (1 , 3 , 4 , 2) T 4 A point set with 4 triangulations the GKZ vectors and the Secondary polytope.

Secondary Polytopes Definition The operation of switching from one triangulation to another is called bistellar flip . Theorem [Gelfand-Kapranov-Zelevinsky] For every point set ❆ of ♥ points in R ❞ corresponds a Secondary polytope ✝✭ ❆ ✮ with dimension ❞✐♠ ✭✝✭ ❆ ✮✮ ❂ ♥ � ❞ � ✶ . The vertices correspond to the regular triangulations of ❆ and the edges to bistellar flips. Secondary polytopes of a pentagon and a quadrilateral.

Enumeration of Triangulations • Two approaches: 1 TOPCOM [Rambau]: combinatorially characterize triangulations 2 Enumeration of Regular Triangulations by Reverse Search [F. Takeuchi, Masada, H.Imai, K.Imai] • Webpage with experiments: http://cgi.di.uoa.gr/ ✘ vfisikop/msc thesis/exper implicit.html

Outline Triangulations 1 Mixed Subdivisions 2 Resultant Polytopes 3

Minkowski Sum Definition The Minkowski sum of two convex polytopes P ✶ and P ✷ is the convex polytope: P ❂ P ✶ ✰ P ✷ ✿❂ ❢ ♣ ✶ ✰ ♣ ✷ ❥ ♣ ✶ ✷ P ✶ ❀ ♣ ✷ ✷ P ✷ ❣ Minkowski sum of a square and a triangle.

Minkowski Cell Let ❆ ✶ ❀ ✿ ✿ ✿ ❀ ❆ ❦ points sets in R ❞ and ❆ ❂ ❆ ✶ ✰ ✿ ✿ ✿ ✰ ❆ ❦ Definition • A subset of ❆ is called Minkowski cell if it can be written as ❋ ✶ ✰ ✁ ✁ ✁ ✰ ❋ ❦ for certain subsets ❋ ✶ ✒ ❆ ✶ ❀ ✿ ✿ ✿ ❀ ❋ ❦ ✒ ❆ ❦ . • A Minkowski cell is fine if all ❋ ✐ are affinely independent and P ❦ ✐ ❂✶ ❞✐♠ ✭ ❋ ✐ ✮ ❂ ❞ . Two fine cells (black, gray) and a non-fine one (white).

Mixed Subdivisions Definition A polyhedral subdivision of ❆ whose cells are all Minkowski cells, is called mixed subdivision . • if all cells are fine is called fine mixed subdivision. • if it is the projection of the lower hull for some lifting on ❆ is called regular mixed subdivision. 3 regular mixed subdivisions and 3 regular fine mixed subdivisions.

The Cayley Trick Definition The Cayley embedding of ❆ ✶ ❀ ✿ ✿ ✿ ❆ ❦ is the point set ❈ ✭ ❆ ✶ ❀ ✿ ✿ ✿ ❀ ❆ ❦ ✮ ❂ ❆ ✶ ✂❢ ❡ ✶ ❣❬ ❆ ✷ ✂❢ ❡ ✷ ❣❬ ✿ ✿ ✿ ❬ ❆ ❦ ✂❢ ❡ ❦ ❣ ✒ R ❞ ✂ R ❦ � ✶ where ❡ ✶ ❀ ❡ ✷ ❀ ✿ ✿ ✿ ❀ ❡ ❦ are an affine basis of R ❦ � ✶ . Proposition (the Cayley trick) polyhedral mixed subdivisions subdivisions of of C ( A 1 , . . . , A k ) A 1 + . . . + A k regular mixed subdivisions regular polyhedral subdivisions regular triangulations regular fine mixed subdivisions

Outline Triangulations 1 Mixed Subdivisions 2 Resultant Polytopes 3

Newton polytopes Definition • The support of a polynomial ❢ ✐ ❂ P ❝ ✐❥ ⑦ � ✦ ❛ ✐❥ is the set: ① s✉♣ ✭ ❢ ✐ ✮ ❂ ❢ ❛ ✐❥ ✷ N ♥ ✐ ✿ ❝ ✐❥ � ✵ ❣ • Given a polynomial ❢ ✐ its Newton polytope ◆ ✭ ❢ ✐ ✮ is the convex hull of its support. f 0 = x 2 y 3 + 3 x − 5 y 2 f 1 = x 5 + y 5 + 3 (0,5) (2,3) (0,2) N ( f 0) N ( f 1) (1,0) (0,0) (5,0)

Resultant Let ❢ ❂ ❢ ✵ ❀ ❢ ✶ ❀ ✿ ✿ ✿ ❀ ❢ ❦ a polynomial system on ❦ variables. Definition • The (sparse) Resultant of ❢ is a polynomial ❘ ✷ ❑ ❬ ❝ ✐❥ ❪ s.t. ❘ ❂ ✵ iff ❢ has a common root. • The Newton polytope of the Resultant polynomial ◆ ✭ ❘ ✮ is called the Resultant polytope . • We call an extreme term of ❘ a monomial which correspond to a vertex of ◆ ✭ ❘ ✮ . Example ❢ ✵ ❂ ❝ ✵✵ ✰ ❝ ✵✶ ① (0 , 1 , 1 , 0) ❢ ✶ ❂ ❝ ✶✵ ✰ ❝ ✶✶ ① (1 , 0 , 0 , 1) R ❂ c 00 c 11 � c 01 c 10

Resultant Extreme Terms Let ❆ ✵ ❀ ❆ ✶ ❀ ✿ ✿ ✿ ❀ ❆ ❦ s.t. ❆ ❥ ❂ ◆ ✭ s✉♣ ✭ ❢ ❥ ✮✮ in Z ❦ , ❆ ❂ ❆ ✵ ✰ ❆ ✶ ✰ ✁ ✁ ✁ ✰ ❆ ❦ Definition A Minkowski cell ✛ is called i -mixed if for all ❥ exists ❋ ❥ ✒ ❆ ❥ s.t. ✛ ❂ ❋ ✵ ✰ ✁ ✁ ✁ ✰ ❋ ✐ � ✶ ✰ ❋ ✐ ✰ ❋ ✐ ✰✶ ✰ ✁ ✁ ✁ ✰ ❋ ❦ where ❥ ❋ ❥ ❥ ❂ ✷ (edges) for all ❥ � ✐ and ❥ ❋ ✐ ❥ ❂ ✶ (vertex). Theorem [Sturmfels] Given a polynomial system ❢ and a regular fine mixed subdivision of the Minkowski sum of the Newton polytopes of its supports we get an extreme term of the resultant ❘ which is equal ❨ ❦ ❨ ❝ ✈♦❧ ✭ ✛ ✮ ❝♦♥st ✁ ✐❋ ✐ ✐ ❂✵ ✛ where ✛ ❂ ❋ ✵ ✰ ❋ ✶ ✰ ✁ ✁ ✁ ✰ ❋ ❦ is an ✐ -mixed cell and ❝♦♥st ✷ ❢� ✶ ❀ ✰✶ ❣ .

An Algorithm Input: A polynomial system ❢ ❂ ❢ ✵ ❀ ❢ ✶ ❀ ✿ ✿ ✿ ❀ ❢ ❦ Output: The Resultant polytope of ❢ : ◆ ✭ ❘ ✮ 1 Compute the Newton polytopes of the supports of ❢ : ❆ ✵ ❀ ✿ ✿ ✿ ❀ ❆ ❦ 2 Compute the Cayley embedding ❈ ✭ ❆ ✵ ❀ ✿ ✿ ✿ ❀ ❆ ❦ ✮ 3 Enumerate all regular triangulations of ❈ ✭ ❆ ✵ ❀ ✿ ✿ ✿ ❀ ❆ ❦ ✮ . i.e. the regular fine mixed subdivisions of ❆ ✵ ✰ ✁ ✁ ✁ ✰ ❆ ❦ 4 For each regular fine mixed subdivision compute the corresponding extreme term of ❘

Mixed Cells Configurations Definition [Michiels, Verschelde] Mixed cells configurations are the equivalence classes of mixed subdivisions with the same ✐ -mixed cells. Ξ Polytope [Michiels, Cools]: Mixed cell configurations as vertices

Resultant Extreme Terms Classification • We care only about mixed cell configurations. • Two regular fine mixed subdivisions of ❆ may produce the same extreme term of ❘ C 01 C 03 C 2 11 C 4 20 2 mixed cell configurations that yield the same extreme term of ❘

Secondary - ☎ - Resultant A 0 A 1 A 2 # mixed subdivisions = 122 # mixed cell configurations = 98 # Resultant extreme terms = 8 mixed cell configurations Resultant extreme terms Secondary Polytope Ξ Polytope Resultant Polytope

A 1 A 2 A 3 An example 0 3 1 5 2 4

Cubical Flips General position assumption: every two faces with the same dimension from two different ❆ ✐ are not parallel. Definition • An affine cube is the Minkowski sum of nonparallel edges and vertices. • Consider the set of cells that are changed by the flip. If the lifted points of this set form an affine cube then the flip is called cubical . Theorem [Sturmfels] Two regular fine mixed subdivisions produce the same extreme term of ❘ iff they are connected by a sequence of non-cubical flips. cub non-cub non-cub

A Combinatorial Test for Cubical Flips • Let ❙ a regular fine mixed subdivision then ❙ has a cubical flip iff exists a set ❈ ✒ ❙ of ✐ -mixed cells s.t. ❈ ❂ ❋ ✶ ✰ ❋ ✷ ✰ ✁ ✁ ✁ ✰ ❋ ❦ and ✽ ❃ ❈ ✶ ❂ ❛ ✶ ✰ ❋ ✷ ✰ ✁ ✁ ✁ ✰ ❋ ✐ ✰ ✁ ✁ ✁ ✰ ❋ ❦ � ✶ ✰ ❋ ❦ ❃ ❃ ❃ ❃ ✿ ❃ ✿ ❃ ❃ ✿ ❁ ❈ ❂ ❈ ✐ ❂ ❋ ✶ ✰ ❋ ✷ ✰ ✿ ✿ ✿ ✰ ❛ ✐ ✰ ✁ ✁ ✁ ✰ ❋ ❦ � ✶ ✰ ❋ ❦ ❃ ❃ ❃ ✿ ❃ ✿ ❃ ✿ ❃ ❃ ❃ ✿ ❈ ❦ ❂ ❋ ✶ ✰ ❋ ✷ ✰ ✁ ✁ ✁ ✰ ❋ ✐ ✰ ✁ ✁ ✁ ✰ ❋ ❦ � ✶ ✰ ❛ ❦ where ❛ ✐ ✷ ❋ ✐ ❀ ❥ ❛ ✐ ❥ ❂ ✶ ❀ ❥ ❋ ✐ ❥ ❂ ✷ . • The cubical flip supported on ❈ of a subdivision ❙ to a subdivision ❙ ✵ consist of changing every ❛ ✐ with ❋ ✐ � ❢ ❛ ✐ ❣ .