Decomposing the Secondary Cayley Polytope - CiteSeerX

Decomposing the Secondary Cayley Polytope Tom Michiels

Ronald Cools

Report TW 281, August 1998

n

Katholieke Universiteit Leuven Department of Computer Science

Celestijnenlaan 200A { B-3001 Heverlee (Belgium)

Decomposing the Secondary Cayley Polytope Tom Michiels

Ronald Cools

Report TW 281, August 1998

Department of Computer Science, K.U.Leuven

Abstract

The vertices of the secondary polytope of a point con guration correspond to its regular triangulations. The Cayley trick links triangulations of one point con guration, called the Cayley polytope, to the ne mixed subdivisions of a tuple of point con gurations. In this paper we investigate the secondary polytope of this Cayley polytope. Its vertices correspond to all regular mixed subdivisions of a tuple of point con gurations. We demonstrate that it equals the Minkowski sum of polytopes, which we call mixed secondary polytopes, whose vertices correspond to regular-cell con gurations.

Keywords : point con guration, polytope, mixed-cell con guration, secondary polytope, bistellar ip. AMS(MOS) Classi cation : 14Q99, 52A39, 52B99.

DECOMPOSING THE SECONDARY CAYLEY POLYTOPE TOM MICHIELS AND RONALD COOLS Abstract. The vertices of the secondary polytope of a point con guration correspond to its regular triangulations. The Cayley trick links triangulations of one point con guration, called the Cayley polytope, to the ne mixed subdivisions of a tuple of point con gurations. In this paper we investigate the secondary polytope of this Cayley polytope. Its vertices correspond to all regular mixed subdivisions of a tuple of point con gurations. We demonstrate that it equals the Minkowski sum of polytopes, which we call mixed secondary polytopes, whose vertices 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 continuation methods for solving polynomial systems (See [HS95] and [VVC94]). The vertices of the secondary polytope (See [GKZ94]) are in a one-to-one correspondence to regular triangulations of a point con guration. We show that for mixed subdivisions, this secondary polytope has a degenerate structure: it can be Minkowski decomposed. In Sections 2 and 3 a brief introduction is given to regular triangulations and regular mixed subdivisions stressing the properties that are important to us. We de ne the I -mixed secondary polytopes in Section 4 and proof in Section 5 that the sum of these I -mixed secondary polytopes is the secondary polytope. The connection between bistellar ips and the edges of the I -mixed secondary polytopes is explained in Section 6. We conclude with an example in Section 7. 2. Preliminaries on Triangulations A triangulation T of a nite point con gurationSA Rd is a collection of cells I A of cardinality d + 1 and dim(conv(I )) = d such that I 2T 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 there exists a lifting vector ! 2 RA such that f(a; ! (a)) j a 2 I g with I 2 T are the upper convex hull faces of f(a; ! (a)) j a 2 Ag. (See [Lee91] for more on regular triangulations.) A circuit is an anely dependent point con guration Z with all proper subsets of Z anely independent. ConsequentlyPthere exists, upPto real multiple, a unique ane relation between the points of the circuit: z2Z z z = 0; z2Z z = 1. (See [Zie95].) In this paper we x j xj = vol(Z n fxg), where vol denotes a translation-invariant volume scaled to be 1 for the unit-simplex. Date : August 17, 1998. This research was conducted as part of project G.0261.96 (Counting and computing all isolated solutions of systems of nonlinear equations) funded by the Fund for Scienti c Research{Flanders (F.W.O.{Vlaanderen) Belgium. 1

2

TOM MICHIELS AND RONALD COOLS

A 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 faces of 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 then 9j : 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 having a 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 then vol(conv(I (1))) = z : vol(conv(I (2))) z Proof. For notational convenience, a set of points is denoted by a (d + 1) (d + 1)-matrix whose rows are the ane coordinates of these points. Since the volume of a unit simplex is scaled to 1, (1)

(2)

F : Z (1) Where Z (1) represent the points in Z n fz(1)g. The points of I (1) are anely independent and thus there exists an orthonormal transformation U such that F G H U: Z (1) = 0 Z 0(1) vol(conv(I (1))) = jdet(I (1))j = det

where Z 0 (1) is a square matrix with the same number of rows as Z (1). Then vol(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 j. Since Z is a circuit and thus anely dependent, applying the same U to I (2) gives (1)

U: ZF(2)

= G0 ZH0 (2) :

Combining all this proves vol(conv(I (1))) = det(I (1)) = det(Z 0 (1)) = z vol(conv(I (2))) det(I (2)) det(Z 0 (2)) z Note that if F = ; then it follows immediately that vol(conv(I (1))) = det(Z (1)) = z : vol(conv(I (2))) det(Z (2)) z

(2)

(1)

(1)

(2)

:

2

Secondary polytopes were introduced in [GKZ94]. The vertices of a secondary polytope correspond 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

3

are 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 guration A Rd is1 X 'T : A ! R : a 7! vol(conv(I )) I ja2I

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 Rd be 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 Subdivisions A (regular) mixed subdivision S of (A1; A2; : S : : ; An) with Ai Rd is a collection of cells C = (C1S; C2; : : : ; Cn) with Ci Ai such that the Ci fe(i 1)g make a (regular) triangulation of Ai 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, Proposition 1.7, page 274],[VGC96, Proposition 3.9]. All properties of (regular) triangulations can be formulated for (regular) mixed subdivisions. S A tuple Z = (Z1; Z2; : : : ; Zn) is a mixed circuit if Zi fe(i 1)g is a circuit. PWe denote the ane coecients of this (mixed) circuit by i;z for z 2 Zi . Note that since 8i : z2Z i;z = 0, 8i : #Zi 6= 1.SA mixed subdivision S is ipable over Z if its corresponding triangulation is

ipable over Zi fe(i 1) g. Two mixed subdivisions are connected by aSbistellar ip ipZ (i 1) g. if their corresponding triangulations are connected by a bistellar ip over Pn Zi fe A cell C of S is involved in a bistellar ip ipZ if C 2= ip Z (S ), i.e., if i=1 #(Zi n Ci) = 1. De nition 3.1. For a mixed subdivision S the characteristic function is 'S = ('S;1; 'S;2; : : : ; 'S;n) with [ X 'S;i : Ai ! R : a 7! vol conv Cj fe(j 1) g : i

C ja2C

i

For notational convenience from now on we will denote the volume of the convex hull of the simplex corresponding to a cell brie y with vol(C ). Clearly 'S 'T for a triangulation T corresponding to a mixed subdivision S . 1

With

P

j we mean the summation over all X for which Y holds.

X Y

4


De 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 subdivisions. 4. Decomposing the Secondary Cayley Polytope In 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 X 'IS;i : Ai ! R : x 7! vol(C ):

Theorem 4.3. 'S;i = Proof.

C 2S jB(C)[fig=I;x2C

i

P

I I f1;2;:::;ng 'S ;i

'S;i (x) = = =

X

vol(C )

C 2Sjx2C X

i

X

I f1;2;:::;ng C 2Sjx2C ^B(C)=I i

X

vol(C )

X

I f1;2;:::;ngji2I C 2Sjx2C ^B(C)[fig=I X 'IS;i(x) I f1;2;:::;ngji2I X 'IS;i (x) I f1;2;:::;ng

vol(C )

i

= =

2

De 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) = f0gk +k +:::+k (Ai) f0gk +k +:::+k where kj = #Aj . 1

2

i

1

i+1

i+2

n


5

Theorem 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 (1)

(A1; A2; : : : ; An ) =

X

I f1;2;:::;ng

I (A1; A2; : : : ; An):

Note that the right inclusion in (1) follows from Theorem 4.3. We will postpone the proof of 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 cellPC of a mixed subdivision S corresponds to a (n + d 1)-dimensional simplex, we have #Ci = n + d and 8i : #Ci 1, thus #B (C ) d. If #I > d + 1 then there is no cell C 2 S such that B (C ) [ fig = I . 2

5. Mixed Secondary Polytopes versus Bistellar Flips. In this section we will focus on the dierence between the characteristic functions of two neighbouring 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 : Fk(j) \ Zk = ; such that de cells of S (1) and S (2) involved in the bistellar ip ipZ can be written as (2)

C (j;k;z) = (Z1 [ F1(j); Z2 [ F2(j); : : : ; Zk n fzg [ Fk(j); : : : ; Zn [ Fn(j))

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 can write the volume of (2) as vol(C (j;k;z)) = f (j) :j k;zj where f (j) is a constant independent of z. Using these notations we formulate the following theorem:

Theorem 5.1. Given a mixed subdivision S (1) and a neighbour S (2) = ipZ (S (1)) then (3)

'S

(1)

;i (x)

'S ;i(x) = i;x : (2)

s X j=1

f (j):

6


Proof. We only need the volumes of cells involved in the bistellar ip to express the dierence between the characteristic function:

'S

(1)

;i (x) 'S ;i (x) = (2)

n s X X

X

j=1 k=1 z2Z jx2C ( z) n s X X X j;k;

k

=

sign( k;z ):vol(C (j;i;z))

i

j

=

i

i

j

k

|

(5)

=

=

f (j): i;z

j=1 k=1ji6=k z2Z jx2F ( ) [Z z2Z jx2F ( ) [Z nfzg s n X X X X (j)

i;z f :

k;z + j=1 k=1ji6=k z2Z jx2F ( ) [Z z2Z jx2F ( ) [Z nfzg k

(4)

X

f (j): k;z +

s X j=1 s X j=1

=0

f (j):

i;x:

=

z2Zijx2Fi(j) {z |

X

z2Zijz6=x2Zi

s X j=1

i

=0

X

f (j):

{z

i;z + }

i

}

X

z2Zi jx2Zi nfzg

j

i

i

i

i

i;z

j

i

i

i;z

f (j) where i;x = 0 for x 2= Zi:

The simpli cations of (4) and (5) are based on the observation that P

P z2Zk

k;z = 0.

2

Observe that the factor sj=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 X 'IS ;i(x) 'IS ;i(x) = f (j) :( i;x): (2)

(1)

j jI=B(F ( ) )[B(Z) that have in uence on 'IS (1) ;i (x) 'IS (2) ;i (x) are those that are involved the restrictions of De nition 4.2, i.e., B (C (j;k;z) ) [ fig = I . Note that j

Proof. The only cells in the ip, and obey B(C (j;k;z)) is independent of the choice of z. Hence we can make the same simpli cations as done in Equations (4) and (5) of the proof of Theorem 5.1 :

'IS ;i(x) 'IS (1)

(2)

;i (x) =

X

f (j) :( i;x):

j 2f1;:::;sgjB(C ( ) )[fig=I B(C (j;i;z)) [ fig = B(Z ) [ B(F (j)), j;i;z

Since 8k : #Zk 6= 1, from (2) follows and this completes the proof. 2 Corollary 5.3. Given a mixed subdivision S (1) and a neighbour S (2) = ipZ (S (1)) then for all I f1; 2; : : : ; ng there exists a cI 2 [0; 1] such that 'IS 'IS = cI ('S 'S ) : (1)

(2)

(1)

(2)


Furthermore

X

I f1;:::;ng

7

cI = 1:

Proof. This follows directly form Theorem 4.3, Theorem 5.1 and Theorem 5.2.

2

We can now prove Theorem 4.6. Proof. We only need to prove the right inclusion . We will show that every vertex f = P 'IS of P PI (A1; A2; : : : ; An) belongs to (A1; A2; : : : ; An). f isPa vertex of I (A1; A2; : : : ; An) maximizing the inproduct < ; v > for some vector v on 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 neighbouring regular mixed subdivisions S (I) = S (I;1); S (I;2); : : : ; S (I;s ) = S such that

'S ; v > 0 for j = 1; 2; : : : ; sI : 'S Applying Theorem 5.3 gives

I 'IS ; v 0 for j = 1; 2; : : : ; sI 'S and thus

I (6) 'S ; v 'IS ; v : Summing (6) over all I 's gives (I )

I

(I ;j +1)

(I ;j )

(I ;j +1)

(I ;j )

(I )

X

'S ; v which proves that f 2 (A1; A2; : : : ; An ).

'IS ; v = f ; v

(I )

2

6. 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 play a 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 set S I := fCI = (Ck ; Ck ; : : : ; Ck ) j C 2 S and B(C ) = I = fk1; k2; : : : ; klgg of 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) =

ip Z (S (1)) and I f1; : : : ; ng with #I d, then (7) S (1)I = S (2)I () 'IS = 'IS : 1

2

l

(1)

(2)

8


Proof. We denote the cells involved in the bistellar ip ipZ between S (1) and S (2) by C (j;i;z) as in (2). (1) S (1)I 6= S (2)I , (2) , (3) , (4) , (5) ,

9i; j; z 2 Zi : i;z > 0; B(C (j;i;z)) = I and CI(j;i;z) 2= S (2)I 9i; j; z 2 Zi : i;z > 0; B(C (j;i;z)) = I and i 2 I 9j : B(F (j) ) [ B(Z ) = I and 9i : #(Zi [ Fi(j)) > 2 9j : B(F (j) ) [ B(Z ) = I 6 'IS 'IS = 1. A dierence between S (1)I and S (2)I can only be caused by a cell C (j;i;z) involved in the bistellar ip between S (1) and S (2) . 2. Since a bistellar ip only aects Ci(j;i;z) we have CI(j;i;z) 2= S (2)I , i 2 I . 3. This follows from 8i : #Zi = 6 1. (j) 4. If B (F ) [ B (Z ) = I then there is always an i such that #(Zi [ Fi(j) ) > 2 because P #(Zi [ Fi(j) ) = n + d + 1, 8i : #(Zi [ Fi(j) ) 1 and the number of i's for which #(Zi [ Fi(j) ) > 1 is smaller than #I ( d). 2 5. This follows from Theorem 5.2. (1)

(2)

Theorem 6.3. The set of regular mixed subdivisions fS (1); S (2); : : : ; S (r)g whose I -mixedcell con gurations are equal for a given I f1; 2; : : : ; ng, i.e. S (1)I = S (2)I = = S (r)I is

interconnected by bistellar ips. Proof. Consider the union of normal cones on the vertices 'S ; 'S ; : : : ; 'S . This union is the set of lifting vectors inducing one I -mixed-cell con guration. One can in an ad-hoc -way describe this set as the solution of a system of homogeneous linear inequalities. Consequently this 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. 2 Theorem 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 = 'IS = = 'IS , then the vertices 'S ; 'S ; : : : ; 'S are interconnected by edges. Proof. This follows directly from Theorem 4.6 and basic properties of Minkowski sums. 2 Theorem 6.3 and 6.4 allows us to generalise Theorem 6.2 for non-neighbouring regular mixed-cell con gurations. Theorem 6.5. Given two regular mixed subdivisions S (1), S (2) and I f1; : : : ; ng with #I d, then (8) S (1)I = S (2)I () 'IS = 'IS : (1)

(2)

(r )

(1)

(1)

(r )

(2)

(2)

(r )

(1)

(2)

Proof. If S (1)I = S (2)I then Theorem 6.3 ensures that one can construct a path of regular mixed-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 other hand, if 'IS = 'IS then Theorem 6.4 ensures that one can construct a path of regular (1)

(2)


9

mixed-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. 2 Theorem 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. 2 7. An Example Consider 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 secondary polytopes. Note that f1g; f2g; f3g (Figures 2, 3 and 4) are singletons corresponding to the 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 gurationsPS f1;2g, S f1;3g and S f2;3g. The mixed-cell con gurations are depicted by drawing them as i2I Ci. References [BS92] [dL95]

L.J. Billera and B. Sturmfels. Fiber polytopes. Ann. of Math., 135(3):527{549, 1992. J.A. de Loera. Triangulations of Polytopes and Computational Algebra. PhD thesis, Cornell University, 1995. [dLHSS96] J.A. de Loera, S. Hosten, F. Santos, and B. Sturmfels. The polytope of all triangulations of a point con guration. Documenta Mathematica, 1:103{119, 1996. Available from http://www.math.uiuc.edu/documenta/. [GKZ94] I.M. Gel'fand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhauser, 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. Beitrage zur Algebra und Geometrie/Contributions to Algebra and Geometry, 37(2):337{346, 1996. Available from http://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 DIMACS Series, pages 443{456. AMS, Providence, R.I., 1991. [LW98] T.Y. Li and X. Wang. On multivariate Descartes' rule { a counterexample. Beitrage zur Algebra und Geometrie/Contributions to Algebra and Geometry, 39(1):1{5, 1998. Available from http://cdns.emis.de/journals/BAG/. [MIK96] T. Masada, H. Imai, and Imai K. Enumeration of regular triangulations. In Proceedings of the Twelfth Annual Symposium on Computational Geometry, pages 224{233. ACM, 1996. [MV] T. Michiels and J. Verschelde. Enumerating regular mixed-cell con gurations. Accepted for publication 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 arbitrary con 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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 applied to 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. In L.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 solving sparse polynomial systems. SIAM J. Numer. Anal., 31(3):915{930, 1994.


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

11

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

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

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

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

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

Figure 5. f1;2g(A1 ; A2; A3)

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

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

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

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

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

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

Figure 7. f2;3g(A1 ; A2; A3) [Zie95]

Figure 8. f1;2;3g(A1; A2; A3)

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] E-mail address, R. Cools: [email protected]