Decomposing the Secondary Cayley Polytope

Decomposing the SecondaryCayley PolytopeTom Michiels Ronald CoolsReport TW281, August 1998

n Katholieke Universiteit LeuvenDepartment of Computer ScienceCelestijnenlaan 200A { B-3001 Heverlee (Belgium)

Decomposing the SecondaryCayley PolytopeTom Michiels Ronald CoolsReport TW281, August 1998Department of Computer Science, K.U.LeuvenAbstractThe vertices of the secondary polytope of a point con�guration corre-spond to its regular triangulations. The Cayley trick links triangulations ofone point con�guration, called the Cayley polytope, to the �ne mixed sub-divisions of a tuple of point con�gurations. In this paper we investigate thesecondary polytope of this Cayley polytope. Its vertices correspond to allregular mixed subdivisions of a tuple of point con�gurations. We demon-strate that it equals the Minkowski sum of polytopes, which we call mixedsecondary polytopes, whose vertices correspond to regular-cell con�gurations.Keywords : point con�guration, polytope, mixed-cell con�guration, sec-ondary polytope, bistellar ip.AMS(MOS) Classi�cation : 14Q99, 52A39, 52B99.

DECOMPOSING THE SECONDARY CAYLEY POLYTOPETOM MICHIELS AND RONALD COOLSAbstract. The vertices of the secondary polytope of a point con�guration correspond to itsregular triangulations. The Cayley trick links triangulations of one point con�guration, calledthe Cayley polytope, to the �ne mixed subdivisions of a tuple of point con�gurations. In thispaper we investigate the secondary polytope of this Cayley polytope. Its vertices correspondto all regular mixed subdivisions of a tuple of point con�gurations. We demonstrate thatit equals the Minkowski sum of polytopes, which we call mixed secondary polytopes, whosevertices correspond to regular-cell con�gurations.1. Introduction.Regular triangulations and mixed subdivisions play an important role in algebraic geometry(See [Stu94a],[Stu94b], [IR96],[LW98] and [Vir84]) and are used in homotopy continuationmethods for solving polynomial systems (See [HS95] and [VVC94]). The vertices of thesecondary polytope (See [GKZ94]) are in a one-to-one correspondence to regular triangulationsof a point con�guration. We show that for mixed subdivisions, this secondary polytope hasa degenerate structure: it can be Minkowski decomposed.In Sections 2 and 3 a brief introduction is given to regular triangulations and regularmixed subdivisions stressing the properties that are important to us. We de�ne the I-mixedsecondary polytopes in Section 4 and proof in Section 5 that the sum of these I-mixedsecondary polytopes is the secondary polytope. The connection between bistellar ips andthe edges of the I-mixed secondary polytopes is explained in Section 6. We conclude with anexample in Section 7. 2. Preliminaries on TriangulationsA triangulation T of a �nite point con�guration A � Rd is a collection of cells I � A ofcardinality d+ 1 and dim(conv(I)) = d such that SI2T conv(I) = conv(A) and 8I; J 2 T :conv(I \ J) = conv(I) \ conv(J). A triangulation T is called regular (or coherent) if thereexists a lifting vector ! 2 RA such that f(a; !(a)) j a 2 Ig with I 2 T are the upper convexhull faces of f(a; !(a)) j a 2 Ag. (See [Lee91] for more on regular triangulations.)A circuit is an a�nely dependent point con�guration Z with all proper subsets of Z a�nelyindependent. Consequently there exists, up to real multiple, a unique a�ne relation betweenthe points of the circuit: Pz2Z zz = 0;Pz2Z z = 1. (See [Zie95].) In this paper we �xj xj = vol(Z n fxg), where vol denotes a translation-invariant volume scaled to be 1 for theunit-simplex.Date: August 17, 1998.This research was conducted as part of project G.0261.96 (Counting and computing all isolated solutionsof systems of nonlinear equations) funded by the Fund for Scienti�c Research{Flanders (F.W.O.{Vlaanderen)Belgium. 1

2 TOM MICHIELS AND RONALD COOLSA circuit has two triangulations T+ = fZ n fzg j z > 0g and T� = fZ n fzg j z < 0g.A triangulation T of A � Rd is called ipable over Z � A if the simplices of T+(�) are facesof simplices of T and if 8I 2 T ; 8J 2 T+(�) : (J � I) ) (8K 2 T+(�) : K [ (I n J) 2 T ).This implies that 9F (1); F (2); : : : ; F (s) such that 8I 2 T : if 9I+ 2 T+ with I+ � I then9j : I = I+ [ F (j) and 8j 2 f1; : : : ; sg; 8I+ 2 T+ : F (j) [ I+ 2 T .De�nition 2.1. Given a circuit Z and a triangulation T ipable over Z then the bistellar ip of T over Z, ipZ(T ), is the triangulation obtained by replacing all cells I of T havinga cell I+ 2 T+ as a face of any dimension by (I n I+) [ I� with I� 2 T�.A cell I of T is involved in a bistellar ip ipZ if I =2 ipZ(T ), i.e., if #(Z n I) = 1.Theorem 2.2. Given a circuit Z and two cells I(1) = Z nfz(1)g[F and I(2) = Z nfz(2)g[F ,not necessarily belonging to the same triangulation thenvol(conv(I(1)))vol(conv(I(2))) = ���� z(1) z(2) ���� :Proof. For notational convenience, a set of points is denoted by a (d + 1) � (d + 1)-matrixwhose rows are the a�ne coordinates of these points. Since the volume of a unit simplex isscaled to 1, vol(conv(I(1))) = jdet(I(1))j = ����det� FZ(1) ����� :Where Z(1) represent the points in Z n fz(1)g. The points of I(1) are a�nely independent andthus there exists an orthonormal transformation U such thatU:� FZ(1) � = � G H0 Z 0(1) �where Z0(1) is a square matrix with the same number of rows as Z(1). Thenvol(conv(I(1))) = jdet(G)j:jdet(Z 0(1))j:Observe that ��Z 0(1)�� is the volume of Z(1) in its own dimension, i.e. j z(1)j. Since Z is a circuitand thus a�nely dependent, applying the same U to I(2) givesU:� FZ(2) � = � G H0 Z0(2) � :Combining all this provesvol(conv(I(1)))vol(conv(I(2))) = �����det(I(1))det(I(2))����� = �����det(Z 0(1))det(Z 0(2)) ����� = ���� z(1) z(2) ���� :Note that if F = ; then it follows immediately thatvol(conv(I(1)))vol(conv(I(2))) = �����det(Z(1))det(Z(2)) ����� = ���� z(1) z(2) ���� : 2Secondary polytopes were introduced in [GKZ94]. The vertices of a secondary polytopecorrespond to the regular triangulations of a polytope. This property was used in [dL95, TI97,MIK96] to enumerate all regular triangulations of point con�gurations. Secondary polytopes

DECOMPOSING THE SECONDARY CAYLEY POLYTOPE 3are generalized by �ber polytopes (See [BS92] and [Zie95, Lecture 9]) and universal polytopes(See [dLHSS96]).De�nition 2.3. The characteristic function of a triangulation T of a point con�gurationA � Rd is1 'T : A! R : a 7! XIja2I vol(conv(I))where vol is a translation-invariant volume scaled to be 1 for the unit simplex.Note that characteristic functions can be regarded as #A-dimensional vectors.De�nition 2.4. The secondary polytope of a point con�guration A � Rd is�(A) := conv (f'T j T a triangulation of Ag) :Theorem 2.5 (Theorem 1.7, page 221 and Theorem 2.11, page 233 [GKZ94]). Let A � Rdbe a point con�guration then:1. the vertices of �(A) correspond to regular triangulations of A, i.e., the normal cones of�(A) are exactly the cones of lifting vectors inducing the regular triangulations;2. the edges of �(A) correspond to the bistellar ips between regular triangulations.3. Preliminaries on Mixed SubdivisionsA (regular) mixed subdivision S of (A1; A2; : : : ; An) with Ai � Rd is a collection of cells C =(C1; C2; : : : ; Cn) with Ci � Ai such that the SCi � fe(i�1)g make a (regular) triangulationof SAi � fe(i�1)g. This de�nition is equivalent with the usual de�nitions [Stu94b, GKZ94]of �ne mixed subdivisions due to the Cayley trick [Stu94a, Lemma 5.2], [GKZ94, Proposition1.7, page 274],[VGC96, Proposition 3.9]. All properties of (regular) triangulations can beformulated for (regular) mixed subdivisions.A tuple Z = (Z1; Z2; : : : ; Zn) is a mixed circuit if SZi�fe(i�1)g is a circuit. We denote thea�ne coe�cients of this (mixed) circuit by i;z for z 2 Zi. Note that since 8i :Pz2Zi i;z = 0,8i : #Zi 6= 1. A mixed subdivision S is ipable over Z if its corresponding triangulation is ipable over SZi � fe(i�1)g. Two mixed subdivisions are connected by a bistellar ip ipZif their corresponding triangulations are connected by a bistellar ip over SZi � fe(i�1)g.A cell C of S is involved in a bistellar ip ipZ if C =2 ipZ(S), i.e., if Pni=1#(Zi nCi) = 1.De�nition 3.1. For a mixed subdivision S the characteristic function is'S = ('S;1; 'S;2; : : : ; 'S;n)with 'S;i : Ai ! R : a 7! XCja2Ci vol�conv�[Cj � fe(j�1)g�� :For notational convenience from now on we will denote the volume of the convex hull ofthe simplex corresponding to a cell brie y with vol(C). Clearly 'S � 'T for a triangulationT corresponding to a mixed subdivision S.1With PXjY we mean the summation over all X for which Y holds.

4 TOM MICHIELS AND RONALD COOLSDe�nition 3.2. The secondary polytope of a tuple of point con�gurations (A1; A2; : : : ; An)with Ai � Rd is�(A1; A2; : : : ; An) := conv (f'S j S a mixed subdivision of (A1; A2; : : : ; An)g) :As a consequence of Theorem 2.5 we have:Theorem 3.3. For a tuple of point con�gurations (A1; A2; : : : ; An):1. the vertices of �(A1; A2; : : : ; An) correspond to regular mixed subdivisions;2. the edges of �(A1; A2; : : : ; An) correspond to bistellar ips between regular mixed subdi-visions. 4. Decomposing the Secondary Cayley PolytopeIn this section we de�ne I-mixed secondary polytopes and present our main theorem,stating that their Minkowski sum equals the secondary polytope.De�nition 4.1. The type B(X) of a tuple of point con�gurations (cell or mixed circuit)X = (X1; X2; : : : ; Xn) is B(X) = fi 2 f1; 2; : : : ; ng j #Xi > 1g.De�nition 4.2. For I � f1; 2; : : : ; ng the I-mixed characteristic function is'IS;i : Ai ! R : x 7! XC2SjB(C)[fig=I;x2Ci vol(C):Theorem 4.3. 'S;i =PI�f1;2;::: ;ng 'IS;iProof. 'S;i(x) = XC2Sjx2Ci vol(C)= XI�f1;2;::: ;ng XC2Sjx2Ci^B(C)=I vol(C)= XI�f1;2;::: ;ngji2I XC2Sjx2Ci^B(C)[fig=I vol(C)= XI�f1;2;::: ;ngji2I 'IS;i(x)= XI�f1;2;::: ;ng'IS;i(x) 2De�nition 4.4. The I-mixed secondary polytope of a tuple (A1; A2; : : : ; An) of point con-�gurations with Ai � Rd is�I(A1; A2; : : : ; An) := conv �f'IS j S is a regular mixed subdivision of (A1; A2; : : :An)g� :The following theorem follows directly from De�nition 4.2:Theorem 4.5. �fig(A1; A2; : : : ; An) = f0gk1+k2+:::+ki�1 � �(Ai)� f0gki+1+ki+2+:::+kn wherekj = #Aj.

DECOMPOSING THE SECONDARY CAYLEY POLYTOPE 5Theorem 4.6 (Main Theorem). The secondary polytope of a tuple of point con�gurations(A1; A2; : : : ; An) can be decomposed in I-mixed secondary polytopes�(A1; A2; : : : ; An) = XI�f1;2;::: ;ng�I(A1; A2; : : : ; An):(1)Note that the right inclusion � in (1) follows from Theorem 4.3. We will postpone the proofof the left inclusion to the next section.Theorem 4.7. If I � f1; 2; : : : ; ng, #I > d+ 1 then �I(A1; A2; : : : ; An) = f0g.Proof. Since a cell C of a mixed subdivision S corresponds to a (n + d � 1)-dimensionalsimplex, we have P#Ci = n + d and 8i : #Ci � 1, thus #B(C) � d. If #I > d + 1 thenthere is no cell C 2 S such that B(C) [ fig = I . 25. Mixed Secondary Polytopes versus Bistellar Flips.In this section we will focus on the di�erence between the characteristic functions of twoneighbouring mixed subdivisions S(1) and S(2) = ipZ(S(1)). From what we know of a bistellar ip, there exist tuples of point con�gurations F (1); F (2); : : : ; F (s) with 8k; j : F (j)k \ Zk = ;such that de cells of S(1) and S(2) involved in the bistellar ip ipZ can be written asC(j;k;z) = (Z1 [ F (j)1 ; Z2 [ F (j)2 ; : : : ; Zk n fzg [ F (j)k ; : : : ; Zn [ F (j)n )(2)with z 2 Zk where k;z > 0 for cells of S(1) and k;z < 0 for S(2). Using Theorem 2.2 we canwrite the volume of (2) as vol(C(j;k;z)) = f (j):j k;zjwhere f (j) is a constant independent of z. Using these notations we formulate the followingtheorem:Theorem 5.1. Given a mixed subdivision S(1) and a neighbour S(2) = ipZ(S(1)) then'S(1) ;i(x)� 'S(2) ;i(x) = � i;x: sXj=1 f (j):(3)

6 TOM MICHIELS AND RONALD COOLSProof. We only need the volumes of cells involved in the bistellar ip to express the di�erencebetween the characteristic function:'S(1) ;i(x)� 'S(2) ;i(x) = sXj=1 nXk=1 Xz2Zkjx2C(j;k;z)i sign( k;z):vol(C(j;i;z))= sXj=1 � nXk=1ji 6=k Xz2Zkjx2F (j)i [Zi f (j): k;z + Xz2Zijx2F (j)i [Zinfzgf (j): i;z�= sXj=1 f (j):� nXk=1ji 6=k Xz2Zkjx2F (j)i [Zi k;z| {z }=0 + Xz2Zijx2F (j)i [Zinfzg i;z�(4) = sXj=1 f (j):� Xz2Zijx2F (j)i i;z| {z }=0 + Xz2Zijx2Zinfzg i;z�(5) = sXj=1 f (j): Xz2Zijz6=x2Zi i;z= � i;x: sXj=1 f (j) where i;x = 0 for x =2 Zi:The simpli�cations of (4) and (5) are based on the observation that Pz2Zk k;z = 0. 2Observe that the factor Psj=1 f (j) in (3) does not depend on x and thus, is a constant,scaling the vector .Theorem 5.2. Given a mixed subdivision S(1) and a neighbour S(2) = ipZ(S(1)) then'IS(1) ;i(x)� 'IS(2) ;i(x) = � XjjI=B(F (j))[B(Z)f (j)�:(� i;x):Proof. The only cells that have in uence on 'IS(1) ;i(x)�'IS(2);i(x) are those that are involvedin the ip, and obey the restrictions of De�nition 4.2, i.e., B(C(j;k;z)) [ fig = I . Note thatB(C(j;k;z)) is independent of the choice of z. Hence we can make the same simpli�cations asdone in Equations (4) and (5) of the proof of Theorem 5.1 :'IS(1) ;i(x)� 'IS(2);i(x) = � Xj2f1;::: ;sgjB(C(j;i;z))[fig=I f (j)�:(� i;x):Since 8k : #Zk 6= 1, from (2) follows B(C(j;i;z)) [ fig = B(Z) [ B(F (j)), and this completesthe proof. 2Corollary 5.3. Given a mixed subdivision S(1) and a neighbour S(2) = ipZ(S(1)) then forall I � f1; 2; : : : ; ng there exists a cI 2 [0; 1] such that'IS(1) � 'IS(2) = cI ('S(1) � 'S(2)) :

DECOMPOSING THE SECONDARY CAYLEY POLYTOPE 7Furthermore XI�f1;::: ;ng cI = 1:Proof. This follows directly form Theorem 4.3, Theorem 5.1 and Theorem 5.2. 2We can now prove Theorem 4.6.Proof. We only need to prove the right inclusion �. We will show that every vertexf =P'IS(I) of P�I(A1; A2; : : : ; An) belongs to �(A1; A2; : : : ; An).f is a vertex of P�I(A1; A2; : : : ; An) maximizing the inproduct < � ;v > for some vectorv on P�I(A1; A2; : : : ; An). Let 'S� be the vertex maximizing this inproduct < � ;v > on�(A1; A2; : : : ; An). Using linear programming, for all I one can build a path of neighbouringregular mixed subdivisions S(I) = S(I;1);S(I;2); : : : ;S(I;sI) = S�such that 'S(I;j+1) � 'S(I;j) ;v� > 0 for j = 1; 2; : : : ; sI :Applying Theorem 5.3 gives'IS(I;j+1) � 'IS(I;j) ;v� � 0 for j = 1; 2; : : : ; sIand thus 'IS� ;v� � 'IS(I) ;v�:(6)Summing (6) over all I 's gives'S� ;v� � X'IS(I) ;v� = f ;v�which proves that f 2 �(A1; A2; : : : ; An). 26. Mixed Secondary Polytopes versus Regular Mixed-Cell Configurations.In this section we will see that the vertices and edges of a I-mixed secondary polytope playa similar role as for the secondary polytope.De�nition 6.1. A cell C = (C1; C2; : : : ; Cn) is called I-mixed if I = B(C). A (regular)I-mixed-cell con�guration is the setSI := fCI = (Ck1 ; Ck2; : : : ; Ckl) j C 2 S and B(C) = I = fk1; k2; : : : ; klggof a (regular) mixed subdivision S.This de�nition generalises the de�nition of [MV].Theorem 6.2. Given a circuit Z, two neighbouring regular mixed subdivisions S(1), S(2) = ipZ(S(1)) and I � f1; : : : ; ng with #I � d, thenS(1)I = S(2)I () 'IS(1) = 'IS(2) :(7)

8 TOM MICHIELS AND RONALD COOLSProof. We denote the cells involved in the bistellar ip ipZ between S(1) and S(2) by C(j;i;z)as in (2).S(1)I 6= S(2)I (1), 9i; j; z 2 Zi : i;z > 0; B(C(j;i;z)) = I and C(j;i;z)I =2 S(2)I(2), 9i; j; z 2 Zi : i;z > 0; B(C(j;i;z)) = I and i 2 I(3), 9j : B(F (j)) [ B(Z) = I and 9i : #(Zi [ F (j)i ) > 2(4), 9j : B(F (j)) [ B(Z) = I(5), 'IS(1) 6= 'IS(2)1. A di�erence between S(1)I and S(2)I can only be caused by a cell C(j;i;z) involved in thebistellar ip between S(1) and S(2).2. Since a bistellar ip only a�ects C(j;i;z)i we have C(j;i;z)I =2 S(2)I , i 2 I .3. This follows from 8i : #Zi 6= 1.4. If B(F (j)) [ B(Z) = I then there is always an i such that #(Zi [ F (j)i ) > 2 becauseP#(Zi [ F (j)i ) = n + d + 1, 8i : #(Zi [ F (j)i ) � 1 and the number of i's for which#(Zi [ F (j)i ) > 1 is smaller than #I(� d).5. This follows from Theorem 5.2. 2Theorem 6.3. The set of regular mixed subdivisions fS(1);S(2); : : : ;S(r)g whose I-mixed-cell con�gurations are equal for a given I � f1; 2; : : : ; ng, i.e. S(1)I = S(2)I = � � � = S(r)I isinterconnected by bistellar ips.Proof. Consider the union of normal cones on the vertices 'S(1) ; 'S(2) ; : : : ; 'S(r) . This unionis the set of lifting vectors inducing one I-mixed-cell con�guration. One can in an ad-hoc-waydescribe this set as the solution of a system of homogeneous linear inequalities. Consequentlythis union of normal cones is convex, and thus all normal cones are interconnected by facets.These facets correspond to the bistellar ips between the regular mixed subdivisions. 2Theorem 6.4. Given a set of all regular mixed subdivisions fS(1);S(2); : : : ;S(r)g whose I-mixed-characteristic functions are equal for a given I � f1; 2; : : : ; ng, i.e. 'IS(1) = 'IS(2) =� � � = 'IS(r) , then the vertices 'S(1) ; 'S(2) ; : : : ; 'S(r) are interconnected by edges.Proof. This follows directly from Theorem 4.6 and basic properties of Minkowski sums. 2Theorem 6.3 and 6.4 allows us to generalise Theorem 6.2 for non-neighbouring regularmixed-cell con�gurations.Theorem 6.5. Given two regular mixed subdivisions S(1), S(2) and I � f1; : : : ; ng with#I � d, then S(1)I = S(2)I () 'IS(1) = 'IS(2) :(8)Proof. If S(1)I = S(2)I then Theorem 6.3 ensures that one can construct a path of regularmixed-cell con�gurations from S(1) to S(2) all sharing the same I-mixed-cell con�guration.Using Theorem 6.2 we know that they all have the same characteristic function. At the otherhand, if 'IS(1) = 'IS(2) then Theorem 6.4 ensures that one can construct a path of regular

DECOMPOSING THE SECONDARY CAYLEY POLYTOPE 9mixed-cell con�gurations from S(1) to S(2) all sharing the same I-characteristic function.Using Theorem 6.2 we know that they all have the same I-mixed-cell con�guration. 2Theorem 6.6. For a I � f1; : : : ; ng with #I � d :1. the vertices of �I(A1; A2; : : : ; An) correspond to the I-mixed-cell con�gurations of(A1; A2; : : : ; An);2. the edges of �I(A1; A2; : : : ; An) correspond to bistellar ips involving I-mixed cells.Proof. This follows from Theorem 6.5. 27. An ExampleConsider the following point con�gurations A1 = A2 = f(0; 0); (1; 0); (1; 1)g and A3 =f(0; 0); (1; 0)g. Figure 1 shows the secondary polytope of (A1; A2; A3) with for each vertex,a regular mixed subdivision. Figures 2, 3, 4, 5, 6, 7 and 8 denote the mixed secondarypolytopes. Note that �f1g;�f2g;�f3g (Figures 2, 3 and 4) are singletons corresponding tothe only triangulation of A1; A2 and A3. The vertices of �f1;2g;�f1;3g;�f2;3g (Figures 5,6 and 7) correspond to mixed-cell con�gurations Sf1;2g, Sf1;3g and Sf2;3g. The mixed-cellcon�gurations are depicted by drawing them as Pi2I Ci.References[BS92] L.J. Billera and B. Sturmfels. Fiber polytopes. Ann. of Math., 135(3):527{549, 1992.[dL95] J.A. de Loera. Triangulations of Polytopes and Computational Algebra. PhD thesis, Cornell Uni-versity, 1995.[dLHSS96] J.A. de Loera, S. Hosten, F. Santos, and B. Sturmfels. The polytope of all triangula-tions of a point con�guration. Documenta Mathematica, 1:103{119, 1996. Available fromhttp://www.math.uiuc.edu/documenta/.[GKZ94] I.M. Gel'fand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimen-sional Determinants. Birkh�auser, Boston, 1994.[HS95] B. Huber and B. Sturmfels. A polyhedral method for solving sparse polynomial systems. Math.Comp., 64(212):1541{1555, 1995.[IR96] I. Itenberg and M.-F. Roy. Multivariate Descartes' rule. Beitr�age zur Algebra und Ge-ometrie/Contributions to Algebra and Geometry, 37(2):337{346, 1996. Available fromhttp://cdns.emis.de/journals/BAG/.[Lee91] C.W. Lee. Regular triangulations of convex polytopes. In P. Gritzmann and B. Sturmfels, editors,Applied Geometry and Discrete Mathematics - The Victor Klee Festschrift, volume 4 of DIMACSSeries, pages 443{456. AMS, Providence, R.I., 1991.[LW98] T.Y. Li and X. Wang. On multivariate Descartes' rule { a counterexample. Beitr�age zur Al-gebra und Geometrie/Contributions to Algebra and Geometry, 39(1):1{5, 1998. Available fromhttp://cdns.emis.de/journals/BAG/.[MIK96] T. Masada, H. Imai, and Imai K. Enumeration of regular triangulations. In Proceedings of theTwelfth Annual Symposium on Computational Geometry, pages 224{233. ACM, 1996.[MV] T. Michiels and J. Verschelde. Enumerating regular mixed-cell con�gurations. Accepted for publi-cation in Discrete Comput. Geom.[Stu94a] B. Sturmfels. On the Newton polytope of the resultant. Journal of Algebraic Combinatorics, 3:207{236, 1994.[Stu94b] B. Sturmfels. Viro's theorem for complete intersections. Annali della Scuola Normale di Pisa,21(3):377{386, 1994.[TI97] F. Takeuchi and H. Imai. Enumerating triangulations for products of two simplices and for arbitrarycon�gurations of points. In Computing and Combinatorics: third annual international conference;Proceedings COCOON'97, volume 1276 of Lecture Notes in Computer Science, pages 470{481.Springer-Verlag, 1997.

10 TOM MICHIELS AND RONALD COOLS(4,4,1,1,3,5,4,3)

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(5,2,2,2,5,2,2,5)

(5,3,1,2,4,3,2,5)

(4,2,3,3,5,1,2,5)

(3,5,1,4,2,3,2,5)

(2,5,2,5,2,2,2,5)

(2,4,3,5,3,1,2,5)

(1,4,4,5,3,1,3,4)

(1,5,3,5,2,2,3,4)

(1,3,5,4,4,1,4,3)

(2,2,5,3,5,1,4,3)(1,5,3,3,2,4,5,2)(2,5,2,2,2,5,5,2)

(3,5,1,2,2,5,4,3)

(3,4,2,1,3,5,5,2)

(3,2,4,1,5,3,5,2) (2,2,5,2,5,2,5,2)

(1,3,5,3,4,2,5,2)

Figure 1. �(A1; A2; A3)(1,1,1,0,0,0,0,0)Figure 2.�f1g(A1; A2; A3) (0,0,0,1,1,1,0,0)Figure 3.�f2g(A1; A2; A3) (0,0,0,0,0,0,0,0)Figure 4.�f3g(A1; A2; A3)[VGC96] J. Verschelde, K. Gatermann, and R. Cools. Mixed-volume computation by dynamic lifting appliedto polynomial system solving. Discrete Comput. Geom., 16(1):69{112, 1996.[Vir84] O.Y. Viro. Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7. InL.D. Faddeev and A.A. Mal'cev, editors, Topology, volume 1060 of Lecture Notes in Mathematics,pages 187{200. Springer{Verlag, 1984.[VVC94] J. Verschelde, P. Verlinden, and R. Cools. Homotopies exploiting Newton polytopes for solvingsparse polynomial systems. SIAM J. Numer. Anal., 31(3):915{930, 1994.

DECOMPOSING THE SECONDARY CAYLEY POLYTOPE 11(1,2,0,1,0,2,0,0) (0,2,1,2,0,1,0,0)

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

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

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

(2,0,1,0,2,1,0,0)Figure 5. �f1;2g(A1; A2; A3) (1,1,0,0,0,0,1,2)

(0,1,1,0,0,0,2,1)Figure 6. �f1;3g(A1; A2; A3)(0,0,0,1,1,0,1,2)

(0,0,0,0,1,1,2,1)Figure 7. �f2;3g(A1; A2; A3)(0,0,1,0,1,0,1,0)

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

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

(1,0,0,0,1,0,0,1)Figure 8. �f1;2;3g(A1; A2; A3)[Zie95] G.M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer{Verlag, New York, 1995.Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200 A,B-3001 Heverlee, Belgium.E-mail address, T. Michiels: [email protected] address, R. Cools: [email protected]