A NEW VIEW ON THE CAYLEY TRICK

Transcription

1 A NEW VIEW ON THE CAYLEY TRICK KONRAD-ZUSE-ZENTRUM FÜR INFORMATIONSTECHNIK (ZIB) TAKUSTR. 7 D BERLIN GERMANY rambau@zib.de joint work with: BIRKETT HUBER, FRANCISCO SANTOS, Mathematical Sciences Research Institute, Berkeley, USA; Universidad de Cantabria, Santander, Spain.

2 INTRODUCTION POINT CONFIGURATIONS AND SUBDIVISIONS point configuration: a set of points A, some points perhaps repeated, and its convex hull faces and intersections: 356 < < and 127 intersect improperly subdivisions: {1345, 123, 235, 257} is a subdivision {134, 345, 123, 235, 257} is a refinement 5

3 INTRODUCTION THE MINKOWSKI SUM A A + B For A,B R d, A + B R d is defined as A + B := {a + b : a A,b B}. B Observation: #A 1 A, #B 1 B : #A 1 #B 1 Π M A + B.

4 INTRODUCTION THE CAYLEY EMBEDDING A C(A,B) For A,B R d, C(A,B) R 2 R d is defined as C(A,B) := (e 1 A) (e 2 B). B Observation: #A 1 A, #B 1 B : #A 1 #B 1 Π C C(A,B).

5 INTRODUCTION MIXED CELLS AND MIXED SUBDIVISIONS + = + = + = + = mixed cell: a Minkowski sum of subsets fine mixed cell: a full-dimensional mixed cell that does not properly contain any other full-dimensional mixed cell (fine) mixed subdivision: a subdivision consisting of (only fine) mixed cells

6 INTRODUCTION THE CAYLEY TRICK THEOREM (GELFAND, KAPRANOV, ZELEVINSKY; STURMFELS; HUBER): LET A, B BE POINT CONFIGURATIONS. THEN THERE IS A ONE-TO-ONE CORRESPONDENCE BETWEEN FINE COHERENT MIXED SUBDIV S OF THE MINKOWSKI SUM A + B AND MINIMAL COHERENT SUBDIV S OF THE CAYLEY EMBEDDING C(A,B)

7 INTRODUCTION REFERENCES GELFAND, KAPRANOV & ZELEVINSKY (1991): Discriminants, Resultants and Multidimensional Determinants STURMFELS (1994): On the Newton polytope of the resultant EMIRIS & CANNY (1994): Efficient incremental algorithms for the sparse resultant and the mixed volume HUBER & STURMFELS (1995): A polyhedral method for solving sparse polynomial systems VERSCHELDE, GATERMANN & COOLS (1996): Mixed-volume computation by dynamic lifting applied to polynomial system solving MICHIELS & VERSCHELDE (1997): Enumerating regular mixed-cell configurations

8 THE GEOMETRIC VIEW THE CAYLEY TRICK ONE PICTURE PROOF

9 THE GEOMETRIC VIEW INDUCED SUBDIVISIONS Given π : vertp B, a π-induced subdivision of B is a subdivision that only contains projections of faces of P. refinement poset ω(a,π). tight subdivision := minimal element. Observation: A = vertices of a simplex all subdivisions induced

10 THE GEOMETRIC VIEW THE CAYLEY TRICK REVISITED THEOREM: LET P := vertp A AND Q := vertq B BE PROJECTIONS OF POINT CONFIGURATIONS WITH P Q Π M A + B, P Q Π C C(A,B). THEN THERE IS A ONE-TO-ONE CORRESPONDENCE BETWEEN TIGHT COHERENT Π M -INDUCED SUBDIVISIONS AND TIGHT COHERENT Π C -INDUCED SUBDIVISIONS

11 APPLICATIONS ZONOTOPAL TILINGS Zonotopes are projections of cubes (Minkowski sums of line segments); zonotopal tilings are induced (mixed) subdivisions of zonotopes; The corresponding Cayley embedding is the Lawrence polytope of the zonotope. COROLLARY FOR EVERY ZONOTOPE THERE IS A ONE-TO-ONE CORRESPONDENCE BETWEEN ZONOTOPAL TILINGS AND SUBDIVISIONS OF ITS LAWRENCE POLYTOPE.

12 APPLICATIONS TRIANGULATIONS OF PRODUCTS OF SIMPLICES p q = C( q,..., }{{} q ), p + 1 times q p dim( q + + q ) dimc( q,..., q ). THEOREM (SANTOS) THERE IS A RECURSIVE FORMULA FOR THE NUMBER OF TRIANGULATIONS OF p 2.

13 CONCLUSION REMARKS AND OPEN PROBLEMS New View provides isomorphisms between fiber polytopes. All subdivisions of Lawrence polytopes are lifting subdivisions (SANTOS). new proof of the BOHNE-DRESS-Theorem. The d-cube is the Cayley-embedding of two (d 1)-cubes. some improved bounds for the minimum triangulation (SANTOS). ω(a,π) is not always connected (R. & ZIEGLER). Is ω( p,π) always connected? Is ω(c p,π) always connected? The Cayley Trick provides an isomorphism between ω( p1 pk,π M ) and ω( p1 pk,π C ). Are there further isomorphisms between posets of induced subdivisions?