heuristic methods for computing the minimal multi-homogeneous bézout number

Applied Mathematics and Computation 146 (2003) 237–256

www.elsevier.com/locate/amc

Heuristic methods for computing theminimal multi-homogeneous B�eezout number

Ting Li, Zhenjiang Lin, Fengshan Bai *,1

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China

Abstract

The multi-homogeneous B�eezout number of a polynomial system is the number of

paths that one has to follow in computing all its isolated solutions with continuation

method. Each partition of variables corresponds to a multi-homogeneous B�eezoutnumber. It is a challenging problem to find a partition with minimal multi-homogeneous

B�eezout number. Two heuristic numerical methods for computing the minimal multi-

homogeneous B�eezout number are presented in this paper. Some analysis of computa-

tional complexity are given. Numerical examples show the efficiency of these two

methods.

� 2002 Elsevier Inc. All rights reserved.

Keywords: Multi-homogeneous B�eezout number; Continuation method; System of polynomial

equations

1. Introduction

For a system of polynomial equations one may wish to determine the

number of isolated solutions and then compute all of them. Garcia and

Zangwill [3] and Drexler [1] suggest that homotopy continuation could be used

to find the full set of isolated solutions of polynomial systems. During the last

* Corresponding author.

E-mail address: [email protected] (F. Bai).1 Supported by National Science Foundation of China G19871047 and National Key Basic

Research Special Fund G1998020306.

0096-3003/$ - see front matter � 2002 Elsevier Inc. All rights reserved.

doi:10.1016/S0096-3003(02)00540-4

mail to: [email protected]

238 T. Li et al. / Appl. Math. Comput. 146 (2003) 237–256

two decades, this method has been developed to be a reliable and efficient

numerical algorithm.Consider a polynomial system of equations,

P1ðx1; x2; . . . ; xnÞ ¼ 0;

P2ðx1; x2; . . . ; xnÞ ¼ 0;

..

.

Pnðx1; x2; . . . ; xnÞ ¼ 0;

ð1:1Þ

where X ¼ ðx1; x2; . . . ; xnÞ 2 Cn. Denote P ¼ ðP1; P2; . . . ; PnÞT . The classical ho-

motopy method for polynomial system is based on B�eezout theory. The number

of isolated solutions of (1.1), and hence the number of curves one has to follow

in the continuation, is bounded above by the total degree (i.e., B�eezout number)

TD ¼Qn

i¼1 di, where di is the degree of the ith equation Pi. However, TD is

often far larger than the number of isolated solutions that the system (1.1)

really has. Hence it may waste too much time by following those unnecessarycurves.

Morgan and Sommese [10] propose the multi-homogeneous B�eezout theory.It is shown that the multi-homogeneous B�eezout number also gives an upper

bound for the number of isolated solutions of a polynomial system. Different

way of partitioning the variables produces different multi-homogeneous B�eezoutnumber. It is desired to find a partition whose multi-homogeneous B�eezoutnumber is the smallest among all possible variable partitions. In fact the

minimal multi-homogeneous B�eezout number is usually smaller (sometime evenfar smaller) than B�eezout number TD. Thus smaller number of paths is followed

in the multi-homogeneous homotopy method. Wampler [12] presents an ex-

haustive search method on finding the optimal partition of variables. However,

it only works for small problems, since the computational complexity of the

algorithm grows exponentially as a function of n, the number of variables.

In this paper, two heuristic methods (fission method and assembly method)

are presented. The idea of our methods is aroused by evolution methods [4,9].

All possible partitions of variables in our algorithms can be regarded as apopulation of individuals which undergoes fission or assembly transforma-

tions. These individuals strive for survival: a selection scheme, biased towards

fitter individuals, selects the next generation. After some generations, it gives

an individual that hopefully represents the optimal partition or a reasonable

approximation of it. The computational cost of these two methods are much

lower than that of Wampler�s exhaustive search method. Large amount of

numerical computation shows that the optimal partition can be obtained for

most of the problems by fission method and/or assembly method.The plan of this paper is as follows. In the next section, general ideas and

basic concepts for the minimal multi-homogeneous are introduced. Basic

T. Li et al. / Appl. Math. Comput. 146 (2003) 237–256 239

problems and the method by Wampler [12] are also given. The fission method

is presented in Section 3, and the assembly method is presented in Section 4. InSection 5 a comparison of these two methods with Wampler�s method is given

in several aspects. Numerical results are given in Section 6. Finally, conclusions

and some discussions are given in Section 7.

2. The minimal multi-homogeneous B�eezout number

Consider the polynomial system (1.1). Let X ¼ ðx1; x2; � � � ; xnÞ 2 Cn, and a

partition of it be T ¼ ðxð1Þ; xð2Þ; . . . ; xðmÞÞ, where

xðjÞ ¼ ðxj1 ; xj2 ; . . . ; xjkj Þ; j ¼ 1; 2; . . . ;m:

K ¼ ½k1; k2; . . . ; km� is called the partition vector of T above. The subscripts

clearly should satisfyPm

j¼1 kj ¼ n;m6 n.Assume that the degree of the polynomial PiðxÞ with respect to xðkÞ in (1.1) is

dik, k ¼ 1; 2; . . . ;m. Then the corresponding degree matrix under the partition T

is given by the following:

D ¼

d11 d12 � � � d1md21 d22 � � � d2m... ..

. ...

dn1 dn2 � � � dnm

26664

37775:

The multi-homogeneous B�eezout number of the partition T, denoting by Bm, is

defined as the coefficient of ak11 ak22 � � � akmm in the following polynomial:

/ða1; a2; . . . ; amÞ ¼ ðd11a1 þ d12a2 þ � � � þ d1mamÞ � � � ðdn1a1 þ dn2a2 þ � � �þ dnmamÞ:

It is equivalent to have

Bm ¼X

ði1;i2;...;inÞd1i1d2i2 � � � dnin ; ð2:1Þ

where i1; i2; . . . ; in 2 f1; 2; . . . ;mg, and the integer j ðj ¼ 1; 2; . . . ;mÞ appears inthe index i1; i2; . . . ; in exactly kj times.

Among all possible partitions of variables of a polynomial system, the

minimal multi-homogeneous B�eezout number is denoted by mB. The following

well known example shows the significance of the difference between the

minimal multi-homogeneous B�eezout number and the classical B�eezout number.

Example 2.1. Consider the matrix eigenvalue problem:

Ax ¼ kx; x 2 Cn; k 2 C: ð2:2Þ


One can view it as a polynomial system of x 2 Cn, k 2 C:

Ax ¼ kx;gTx ¼ 1;

�ð2:3Þ

where g 2 Cn is a randomly chosen vector. Clearly, the classical B�eezout number

of system (2.3) is TD ¼ 2n. But we all know that the eigenvalue problem (2.2)

only has n solutions. Let the partition of variables in (2.3) be fx1; . . . ; xng; fkg.The corresponding Bm is exactly n. Hence mB ¼ n for the system (2.3). The

difference between the minimal multi-homogeneous B�eezout number mB ¼ nand the classical B�eezout number TD ¼ 2n is very dramatic, particularly as n is alittle big.

In order to obtain the minimal over all possible Bm, Wampler�s method has

to visit all possible partitions of variables. Hence the following two sub-

problems has to be considered.Go through all possible partitions. The total number of all possible partitions

is Bell number, denoted by BðnÞ. The combinatorics model for Bell number is:

all possible ways of putting n distinct balls into m identical boxes, where

16m6 n. The following recursive relationship is hold for Bell numbers.

BðnÞ ¼Xn�1k¼0

n� 1

k

� �BðkÞ; Bð0Þ ¼ 1:

An estimation is given by [7]

ðn=2Þðn=2Þ < BðnÞ < n!:

This means that Bell number is exponentially increasing as n grows. For ex-

ample, Bð5Þ ¼ 32, Bð10Þ ¼ 115; 975, Bð15Þ ¼ 1; 382; 958; 545. When n is big, it

is impossible to go through so many partitions. That is the reason whyWampler�s method only works for small problems.

Compute the multi-homogeneous B�eezout number for any given partition. Arecursive algorithm is established by Wampler [12]:

bðD;K; iÞ ¼ 1 if i ¼ nþ 1;Pmj¼1;kj 6¼0 dij � bðD;K � ej; iþ 1Þ otherwise

�

and Bm ¼ bðD;K; 1Þ. Here K ¼ ½k1; k2; . . . ; km� is the partitioning vector and D is

the degree matrix, ej ¼ ð0; 0; . . . ; 0; 1; 0; . . . ; 0Þ 2Zn. The computational cost

of such an algorithm is also exponentially increasing as n grows.

In the following sections, we present two efficient heuristic methods to

finding out the partition which corresponds to mB, the minimal multi-homo-geneous B�eezout number, or an approximation of it, without exhaustive

searching for all partitions. To compare the results properly, Wampler�s re-


cursive algorithm mentioned above is adopted to computing Bm for any given

partition.

3. Fission method

Let X ¼ ðx1; x2; . . . ; xnÞ be the unknown vector in system (1.1), and a par-

tition of it be

T ¼ ðxð1Þ; xð2Þ; . . . ; xðmÞÞ;

where xðjÞ ¼ ðxj1 ; xj2 ; . . . ; xjkj Þ, j ¼ 1; 2; . . . ;m. For simplicity, we often denote

the partition T above as fG1;G2; . . . ;Gmg, where Gj ¼ fj1; j2; . . . ; jkjg,j ¼ 1; 2; . . . ;m.

Example 3.1. Suppose X ¼ ðx1; x2; . . . ; x5Þ and T ¼ fxð1Þ; xð2Þ; xð3Þg, where

xð1Þ ¼ fx1; x2g, xð2Þ ¼ fx3; x5g, xð3Þ ¼ fx4g. Then T is denoted by f1; 2g;f3; 5g; f4g.

To eliminate the non-uniqueness, we always assume that

ij < il for all j < l;i1 < j1 for all i < j:

This means that variables inside a group are arranged in an increasing order oftheir subscripts, and the groups in a partition are also placed in an increasing

order of the subscripts of their first variables.

Definition 3.1. T1 ¼ ðxð1Þ1 ; xð2Þ1 ; . . . ; xðmþ1Þ1 Þ is called a partition fissured from the

partition T0 ¼ ðxð1Þ0 ; xð2Þ0 ; . . . ; xðmÞ0 Þ if there exists an index i, ð16 i6mÞ such that

xðjÞ1 ¼ xðjÞ0 8j < i;

xðjþ1Þ1 ¼ xðjÞ0 8j > i;

(j ¼ 1; 2; . . . ;m

and

xðiÞ1 [ xðiþ1Þ1 ¼ xðiÞ0 :

Example 3.2. Consider the following

f1; 2; 3g f4; 5; 6g f7; 8g+ fissure

f1; 2; 3g f4gf5; 6g f7; 8g

Denote T1 ¼ ff1; 2; 3g; f4g; f5; 6g; f7; 8gg, T0 ¼ ff1; 2; 3g; f4; 5; 6g; f7; 8gg. ByDefinition 3.1, T1 is a partition fissured from the partition T0.


Definition 3.2. If T1 is fissured from T0, then T0 is called the parent of T1, and T1is called a child of T0.

It is easy to see that a partition T may have more than one child. We call all

of T�s children its offspring.

Example 3.3. Consider a partition T0 ¼ ff1; 2; 3g; f4; 5gg. Its offspring is the

following set fT11; T12; T13; T14g, where

T11 ¼ f1gf2; 3gf4; 5g; T12 ¼ f1; 2gf3gf4; 5g;T13 ¼ f1; 3gf2gf4; 5g; T14 ¼ f1; 2; 3gf4gf5g:

Assume the partition of variables chosen at step t is denoted by P ðtÞ. One of

the groups in the current partition fissures to generate new partitions. The setof all possible partitions generated from P ðtÞ in this way is called the offspring

of PðtÞ, and is denoted by CðtÞ. The fitness of each element in CðtÞ is evaluatedby computing its multi-homogeneous B�eezout number. A general structure of

the fission algorithm is as follows.

Algorithm 3.1. Fission method

begin

t 1;

initialize P ðtÞ;whileðt < nÞdobegin

fissure P ðtÞ to yield CðtÞ;evaluate CðtÞ;select P ðt þ 1Þ from CðtÞ;t t þ 1;

end

end

The fission method can be thought as a simplified evolution process. Regard

P ðtÞ as a population of individuals at the generation t. Each individual rep-

resents a partition of variables, and it is evaluated by measuring its fitness. One

of the individuals fissures to form new individuals called offspring CðtÞ, arethen evaluated. A new population is formed by selecting the most fit individual

from the offspring population. After several generations, the algorithm con-

verges to one individual, which hopefully represents an optimal or suboptimal

partition of variables.


Remark 3.1. For the initial population of individuals, one natural idea is to use

T ¼ f1; 2; . . . ; ng since T cannot be produced by any partitions through fis-suring operation. Namely, T has no parents. The corresponding Bm under such

partition T is the classical B�eezout number TD.

Remark 3.2. The evaluation function. It is the criterion to determining indi-

viduals� fitness. The most fit individual can be selected from the offspring

population by using the evaluation function. Bm, the multi-homogeneous

B�eezout number of the variable partition, is regarded as the evaluation function.

Remark 3.3. The Criterion to stop. The process of evolution from the initial

population T ¼ f1; 2; . . . ; ng continues until the nth population Tn ¼ f1g;f2g; . . . ; fng, which cannot be fissured any further. Then the process of evo-

lution ends.

Here we adopt the strategy of storing the optimal individual during theprocess of calculation. The optimal partition obtained by fission method is

denoted by fission partition (FP), and the corresponding multi-homogeneous

B�eezout number Bm is denoted by fission B�eezout (FB). Let us survey the pro-

cedure of Fission method calculation through a simple example.

Example 3.4. Consider a polynomial system:

x21 þ x2 þ 1 ¼ 0;

x1x3 þ x2 þ x4 ¼ 0;

x1 þ x2x3 þ x4 ¼ 0;

x1 þ x3x4 þ 4 ¼ 0:

In the following, we denote the present optimal individual by POI. The Fissionmethod runs as follows:

Pð1Þ : T ¼ f1 2 3 4g;

D ¼

2

2

2

2

2664

3775;

Bm ¼ TD ¼ 16;

POI : T ¼ f1; 2; 3; 4g; Bm ¼ 16:


By looking for all possible partitions fissured from T ¼ f1; 2; 3; 4g, it gives
Cð1Þ :
T1 ¼ f1; 2; 3gf4g T2 ¼ f1; 2; 4gf3g T3 ¼ f1; 2gf3; 4g T4 ¼ f1; 3; 4gf2g

D ¼

2 0

2 1

2 1

1 1

2664

3775 D ¼

2 0

1 1

1 1

1 1

2664

3775 D ¼

2 0

1 1

1 1

1 2

2664

3775 D ¼

2 1

2 1

1 1

2 0

2664

3775

Bm ¼ 16 Bm ¼ 6 Bm ¼ 10 Bm ¼ 16

T5 ¼ f1; 3gf2; 4g T6 ¼ f1; 4gf2; 3g T7 ¼ f1gf2; 3; 4g

D ¼

2 1

2 1

1 11 1

2664

3775 D ¼

2 1

1 1

1 21 1

2664

3775 D ¼

2 1

1 1

1 21 2

2664

3775

Bm ¼ 13 Bm ¼ 14 Bm ¼ 16

The set Cð1Þ contains seven elements. By taking one from Cð1Þ whose corre-

sponding Bm is the smallest, the present optimal individual is obtained as fol-

lows:

POI : T ¼ f1; 2; 4gf3g; Bm ¼ 6:

P ð2Þ should be taken from the set C(1), and here it is identical to the present

optimal individual at this step.

Pð2Þ : T ¼ T2 ¼ f1; 2; 4gf3g:

Similarly, by looking for all possible partitions fissured from T ¼ f1; 2; 4gf3g,it gives

Cð2Þ :

T1 ¼ f1gf2; 4gf3g T2 ¼ f1; 2gf3gf4g T3 ¼ f1; 4gf2gf3g

D ¼

2 1 0

1 1 1

1 1 1

1 1 1

2664

3775 D ¼

2 0 0

1 1 1

1 1 1

1 1 1

2664

3775 D ¼

2 1 0

1 1 1

1 1 1

1 0 1

2664

3775

Bm ¼ 12 Bm ¼ 12 Bm ¼ 11


The POI, the present optimal individual at this step is the same as last step,

since there is no element in Cð2Þ whose corresponding Bm could be any smallerthan 6.

POI : T ¼ f1; 2; 4gf3g; Bm ¼ 6:

However P ð3Þ has to be an element from Cð2Þ, whose corresponding Bm is the

smallest over all elements in Cð2Þ. Hence

Pð3Þ : T ¼ T3 ¼ f1; 4gf2gf3g;

Cð3Þ :

T1 ¼ f1gf2gf3gf4g

D ¼

2 1 0 0

1 1 1 1

1 1 1 1

1 0 1 1

2664

3775

Bm ¼ 14

POI : T ¼ f1; 2; 4gf3g; Bm ¼ 6;

Pð4Þ : T ¼ T1 ¼ f1gf2gf3gf4g:

Hence the optimal partition of variables obtained by fission method is

FP ¼ f1; 2; 4gf3g, and the corresponding FB ¼ 6.

For n ¼ 4, the total number of all possible partitions is Bð4Þ ¼ 15. The

minimal multi-homogeneous B�eezout number mB ¼ 6. Adopting fission

method, we calculate 12 partitions to obtain the value of the optimum indi-

vidual which equal to mB. As the number of variables n grows, the percentageof partitions calculated among all possible ones will decrease.

4. Assembly method

An analogous algorithm which basically reverses the procedure of fission

method is presented in this section. It is called assembly method.

Definition 4.1. We say that a partition T1 ¼ ðxð1Þ1 ; xð2Þ1 ; . . . ; xðmÞ1 Þ is assembledfrom

T0 ¼ ðxð1Þ0 ; xð2Þ0 ; . . . ; xðmþ1Þ0 Þ;


if there exists i, j (without loss of generality, we assume that i < j), such

that

xðiÞ1 ¼ xðiÞ0 [ xðjÞ0 ;

xðkÞ1 ¼ xðkÞ0 8k < j and k 6¼ i;

xðkÞ1 ¼ xðkþ1Þ0 8kP j:

8>><>>:

Example 4.1. Consider

f1; 2; 3g f4gf5; 6g f7g+ assemble

f1; 2; 3g f4; 5; 6g f7g

Analogous to fission method, if T1 is assembled from T0, then T0 is called

parent of T1; and T1 is called a child of T0.A general structure of the assembly algorithm is as follows:

Algorithm 4.1. Assembly method

begint 1;

initialize P ðtÞ;whileðt < nÞdobegin

assemble P ðtÞ to yield CðtÞ;evaluate CðtÞ;select P ðt þ 1Þ from CðtÞ;t t þ 1;

end

end

Remark 4.1. In the assembly method, we take T ¼ f1gf2g � � � fng as initial

population since T cannot be generated from any partitions through assem-

bling. The process of evolution from T continues until the nth population

Tn ¼ f1; 2; . . . ; ng, which cannot be assembled. Then the algorithm ends.

Here again, we adopt the strategy of storing the optimal individual in theprocess of calculation. The partition obtained by assembly method is denoted

by assembly partition (AP), and the corresponding Bm is denoted by assembly

B�eezout number (AB).


5. Complexity of algorithms

In Section 2 we point out that in order to obtain the minimal Bm, all possible

partitions of variables have to be gone through by Wampler�s method. The

total number of all possible partitions is the Bell number which grows expo-

nentially.

In the following, we consider the numbers of partitions which have to be

visited by the fission method and assembly method. This is certainly related to

the complexity of our methods.

Lemma 5.1. There are 2n�1 � 1 different possible cases to partitioning a set withn elements into two subsets.

Proof. Deleting all multiplicities, the number of different possible cases to

partitioning a set with n elements into two subsets should always be

1

2fC1

n þ C2n þ � � � þ Cn�1

n g:

Notice that

C0n þ C1

n þ � � � þ Cnn ¼ ð1þ 1Þn ¼ 2n

and thus the fact

C1n þ � � � þ Cn�1

n ¼ 2n � 2:

Hence the conclusion holds. �

Lemma 5.2. Suppose x; y 2N [ f0g, then 2x þ 2y 6 2xþy þ 1.

Proof. When x ¼ 0 or y ¼ 0, the equality holds. Hence the result holds. When

xy 6¼ 0, one can assume xP y, without loss of generality. Hence

2x þ 2y 6 2� 2x 6 2xþy < 2xþy þ 1:

Thus the conclusion holds. �

Lemma 5.3. Suppose n 2N, m 2N and 16m6 n; xi 2N, i ¼ 1; . . . ;m. IfPmi¼1 xi ¼ n, then ð2x1�1 � 1Þ þ � � � þ ð2xm�1 � 1Þ6 2n�m � 1.

Proof. It is proved by induction on the number m.

(1) Let m ¼ 1, then x1 ¼ n. Thus 2x1�1 � 1 ¼ 2n�1 � 1, which proves the result.

(2) Assume that the result of the lemma holds for m ¼ k, that is,

2x1�1 � 1þ � � � þ 2xk�1 � 16 2n�k � 1:


Now consider the case of m ¼ k þ 1. By Lemma 5.2, one has

2x1�1 � 1þ 2x2�1 � 16 2x1þx2�2 � 1 ¼ 2ðx1þx2�1Þ�1 � 1:

By the assumption of induction and the inequality above, one gets

2x1�1 � 1þ � � � þ 2xkþ1�1 � 16 2ðx1þx2�1Þ�1 � 1þ � � � þ 2xkþ1�1 � 1

6 2ðx1þx2�1þx3þ��þxkþ1Þ�k � 1 ¼ 2n�1�k � 1

¼ 2n�ðkþ1Þ � 1:

Hence the result is proved. �

Now we consider the number of partitions used by fission method, referring

Algorithm 3.1 for details.

P(1): T1 ¼ f1; 2; . . . ; ng, the partition that the algorithm starts.

C(1): It is obtained by partitioning T1 into all possible two subsets.

By Lemma 5.1, the number of partitions generated here is 2n�1 � 1.

P(2): T2 ¼ fG1;G2g is chosen from Cð1Þ. Suppose jG1j ¼ x1, jG2j ¼ x2.C(2): Case 1: fG11;G12;G2g, where G11 [ G12 ¼ G1.

By Lemma 5.1, the number of partitions generated in this way is

2x1�1 � 1.

Case 2: fG1;G21;G22g, where G21 [ G22 ¼ G2.

By Lemma 5.1, the number of partitions generated in this way is

2x2�1 � 1.

Hence the number of elements in C(2) is bounded above by 2n�2 � 1, by

Lemma 5.3. Analogously, the number of elements in CðkÞ is bounded above by2n�k � 1.

Thus the total number of partitions used by fission method is

Sumf 6 1þ ð2n�1 � 1Þ þ ð2n�2 � 1Þ þ � � � þ ð2n�ðn�1Þ � 1Þ¼ ð2n�1 þ 2n�2 þ � � � þ 21Þ � nþ 2 ¼ 2n � 2� nþ 2 ¼ 2n � n:

This proves the following result.

Theorem 5.1. Let the number of partitions used in fission method is Sumf . Thenone has Sumf ¼ Oð2nÞ.

The result above shows that the number of partitions visited by fission

method is much less than that of Wampler�s method, but it still grows expo-

nentially as n grows.


Remark 5.1. When each population has a single set fissured from its parent,

the equality above holds. This is the worst case in the computation.

Consider an example as follows.

P(1): T1 ¼ f1; 2; 3; 4g and the number of Cð1Þ is 24�1 � 1;

P(2): T2 ¼ f1; 3; 4gf2g and the number of Cð2Þ is 23�1 � 1 ¼ 24�2 � 1;

P(3): T3 ¼ f1; 3gf2gf4g and the number of Cð3Þ is 22�1 � 1 ¼ 24�3 � 1;

P(4): T4 ¼ f1gf2gf3gf4g.Sumf ¼ 1þ 24�1 � 1þ 24�2 � 1þ 24�3 � 1 ¼ 24 � 4.

For the assembly method, one can easily get the number of partitions used

as follows:

Table

Compa

N

PNm

PNFB

PNA

Suma ¼ 1þ C2n þ C2

n�1 þ � � � þ C22

¼ 1þ nðn� 1Þ2

þ ðn� 1Þðn� 2Þ2

þ � � � þ 1 ¼ Oðn3Þ:

Hence the following result is proved.

Theorem 5.2. The number of partitions that assembly method visits is a poly-nomial function of n.

Notice that the numbers of partitions used by fission method and assembly

method are much less than that of Wampler�s method. In Table 1, N denotes

the number of the unknowns and PNmB, PNFB and PNAB denote the number of

partitions visited by Wampler�s method, fission method and assembly method,

respectively.

The numbers of partitions required in those three methods clearly satisfy

PNmB > PNFB > PNAB. Fission method and assembly method both use less

partitions than Wampler�s method and save time for calculation through suchmodification. The number of partitions computed by assembly algorithm even

grows as a polynomial function of N. But what we care about most is whether

the destination function can reach the optimal value. In the next section, nu-

merical experiments are presented for special examples and also for randomly

generated systems of sparse polynomial equations, respectively.

1

rison of the numbers of partitions

3 4 5 6 7 8 10 12 15

B 5 15 52 203 877 4140 115975 4213597 1382958545

<5 <12 <27 <58 <121 <248 <1014 <4084 <32753

B 5 11 21 36 57 85 166 287 561


6. Numerical results

A large amount of numerical computation is done with the fission method

and the assembly method, which shows the efficiency of these two methods. All

of our computations in this paper are performed on PIII/700, with C as the

programming language.

6.1. Numerical examples

The fission method and the assembly method are first tested with 20 poly-

nomial systems, which are all concrete problems from science and engineering.

We refer the details of those 20 examples to the appendix of [7]. Computational

results are given. Some comparisons are also made with Wampler�s method.

Notations used are as follows:

N the number of unknown

TD total degree, i.e. the classical B�eezout numbermB the minimal multi-homogeneous B�eezout number

FB the fission B�eezout number, the optimal Bm obtained by fission method

AB the assembly B�eezout number, the optimal Bm obtained by assembly

method

Time time spent on calculations

PN the number of partitions calculated in algorithms

Note that ‘‘<0.0001’’ shown in Table 2 means that the time spent for suchcomputation cannot be measured by the computer we use.

Results of Table 2 clearly show that the computing time and the number of

partitions of the fission method and the assembly method are far less than

those of Wampler�s method, as the scale of problems becomes a little big.

All the computational results by our fission method, reach the optimal so-

lution. The fission B�eezout numbers computed for those 20 examples are all

equals to their minimal multi-homogeneous B�eezout numbers. The assembly

B�eezout numbers, which are obtained by the assembly method, differ fromminimal multi-homogeneous B�eezout numbers only for one problem among 20.

For this one, the assembly B�eezout number is not far away from the minimal

multi-homogeneous B�eezout number.

From Table 2, we can see that the number of partitions which has to be

calculated by the assembly method is the smallest. That is also clear from the

analysis in Section 5. However, Table 2 clearly show that the computing time it

consumes is much more than that of the fission method. By carefully com-

paring the calculation procedures of these two methods, one can easily find thereason for this. The fission method uses more partitions. But about half of

them, the size of the corresponding degree matrix is n� 2. The assembly

Table 2

Computation results for 20 examples

N TD mB FB AB

Eq. (1) 2 16 10 10 10

Eq. (2) 4 625 384 384 384

Eq. (3) 4 256 96 96 96

Eq. (4) 4 144 62 62 62

Eq. (5) 4 900 450 450 450

Eq. (6) 5 16 16 16 16

Eq. (7) 6 8 8 8 8

Eq. (8) 8 5764801 645120 645120 645120

Eq. (9) 8 128 16 16 16

Eq. (10) 6 64 20 20 20

Eq. (11) 11 2048 320 320 320

Eq. (12) 8 576 193 193 193

Eq. (13) 7 4608 1361 1361 1361

Eq. (14) 10 3628800 3628800 3628800 3628800

Eq. (15) 9 362880 362880 362880 362880

Eq. (16) 6 1024 216 216 216

Eq. (17) 5 108 56 56 56

Eq. (18) 4 1344 368 368 368

Eq. (19) 10 96 44 44 48*

Eq. (20) 8 256 96 96 96

Time-mB Time-FB Time-AB PNmB PNFB PNAB

Eq. (1) 2 < 0.0001 <0.0001 <0.0001 2 2 2

Eq. (2) 4 <0.0001 <0.0001 <0.0001 15 10 11

Eq. (3) 4 <0.0001 <0.0001 <0.0001 15 10 11

Eq. (4) 4 <0.0001 <0.0001 <0.0001 15 10 11

Eq. (5) 4 <0.0001 <0.0001 <0.0001 15 11 11

Eq. (6) 5 <0.0001 <0.0001 <0.0001 52 26 21

Eq. (7) 6 0.0100 <0.0001 <0.0001 203 46 36

Eq. (8) 8 5.5480 0.1100 0.5800 4140 160 85

Eq. (9) 8 1.8620 0.0710 0.6000 4140 181 85

Eq. (10) 6 0.0100 <0.0001 <0.0001 203 45 36

Eq. (11) 11 1490.53 0.4010 1221.95 678570 1187 221

Eq. (12) 8 0.2200 0.0100 0.1700 4140 169 85

Eq. (13) 7 0.2400 0.0100 0.0500 877 108 57

Eq. (14) 10 6113.00 4.3670 209.86 115975 607 166

Eq. (15) 9 192.37 0.9110 15.022 21147 305 121

Eq. (16) 6 0.0100 <0.0001 <0.0001 203 45 36

Eq. (17) 5 <0.0001 <0.0001 <0.0001 52 26 21

Eq. (18) 4 <0.0001 <0.0001 <0.0001 15 11 11

Eq. (19) 10 2967.01 6.4400 89.8420 115975 693 166

Eq. (20) 8 0.8610 0.0100 0.5700 4140 165 85


method requires less partitions, comparing with the fission method. But for

more than half of all partitions, the size of the corresponding degree matrix is


n� ðn� 1Þ. It is much more expensive to computing Bm for an n� ðn� 1Þdegree matrix. Hence a lot of time has been spent on computing Bm in assemblymethod.

6.2. Further numerical results

In order to carefully compare the efficiency of the fission method, the as-sembly method with Wampler�s method, numerical experiments are done for

sparse polynomial systems which are generated randomly in this section.

Let n denote the number of unknowns, d and s denote the degree and the

sparsity of a polynomial system, respectively. Note that numbers n and d give

number of all possible monomials in a polynomial. Hence the sparsity is the

proportion that monomials appear in the polynomial.

For example, let n ¼ 3, d ¼ 2. Then the set M of all possible monomials is

given by

M ¼ f1; x1; x2; x3; x21; x1x2; x1x3; x22; x2x3; x23g;

where T ¼ 10 gives the number of elements of the set M. Let n ¼ 6, d ¼ 5,

s ¼ 5%, then T ¼ 462, N ¼ 23. That is, each polynomial of the system can

contain not more than 23 monomials.

Some notations used in this section are given as follows.

n the number of unknowns of a polynomial systemd the maximum degree of the polynomial system

s the sparsity of the polynomial system

PS the percentage of success in reaching mB

EB relative error between computing results r and mB, namely

ðr �mBÞ=mB

ET relative error between computing results r and TD, namely

ðTD� rÞ=TDTime-mean the mean of the time spent on calculationPN-mean the mean of the number of partitions used

ETB relative error between total degree TD and multi-homogeneous B�eezoutnumber mB, namely ðTD�mBÞ=TD

Take n ¼ 6, d ¼ 4, s ¼ 5%, and n ¼ 8, d ¼ 3, s ¼ 5%. 400 instances are

computed for each case. Computational results are given in Table 3.

In Table 3, numerical examples indicate that one can obtain the optimal

partition for almost all cases by these two methods. It shows that our twomethods are extraordinary efficient. Note that the assembly method requires

less partitions than the fission method, by the analysis in Section 5 and also

numerical results in Table 3. However the assembly method takes more com-

Table 3

Computation results for random examples

PS (%) Time-mean PN-mean

n ¼ 6, d ¼ 4, s ¼ 5%

Wampler�s method 100 0.0178 203

Fission method 97.25 0.0028 49

Assembly method 96.75 0.0053 36

n ¼ 8, d ¼ 3, s ¼ 5%

Wampler�s method 100 6.4485 4140

Fission method 96.25 0.0692 177

Assembly method 96.75 0.2967 85


puter time, shown by the numerical results. That is because computing the

multi-homogeneous B�eezout number for a partition with more groups is much

more expansive than that for a partition with less groups.

More computation is done for different sparsity. The ratio of success is

plotted in the Fig. 1.

It is shown by Fig. 1 that the percentage that the computational results bythe fission or assembly method reach mB, decreases as the sparsity s decrease.

Fig. 1. The percentage of success to reaching mB.

Table 4

Difference between TD and mB (n ¼ 6, d ¼ 3)

ETB 0 >0 and <0.3 P0.3 and <0.5 P0.5 and <0.7 P0.7

s ¼ 8% 53% 22.5% 10.5% 10% 4%

s ¼ 4% 1% 6.5% 19% 30% 43.5%

Table 5

Distribution of the difference between AB and mB (n ¼ 6, d ¼ 3, s ¼ 4%)

<0.02 P0.02 and

<0.05

P0.05 and

<0.1

P0.1 and

<0.2

P0.2 and

<0.5

P0.5

EB 29.11% 32.91% 15.19% 11.4% 3.8% 7.59%

Table 6

Difference between AB and mB (n ¼ 6, d ¼ 3, s ¼ 4%)

<0.02 <0.05 <0.1 <0.2 <0.5

EB 29.11% 62.02% 77.21% 88.61% 92.41%


It sounds unreasonable. The results shown by Table 4 give a solid answer for

that. As s ¼ 8%, among all 400 polynomial system we computed, 53% of them

make mB ¼ TD, and only 4% of them make mB and TD dramatically differ-

ent. However as the sparsity s ¼ 4%, only 1% of them make mB ¼ TD, while

43.5% of them make mB and TD dramatically different. Hence at the cases of slarger, although one get mB through computation for many cases, it may only

give a trivial, since mB ¼ TD holds at a very high probability. As the sparsity sgoes smaller, one can hope to get more non-trivial results.

As a heuristic method, it is acceptable to have an approximate optimal

solution through the computation. Then the distance between the approximate

solution and the exact solution is what we concern. Some statistics of the non-

success cases for n ¼ 6, d ¼ 3, s ¼ 4% by assembly method is given in Tables 5

and 6.

Statistics in Tables 5 and 6 clearly shows that those computational results

obtained by assembly method which do not equal to mB, are mostly good

approximations of mB. Similar conclusion hold for other case of n and d, and itis also true for fission method.

7. Conclusions and discussions

The fission method and the assembly method are proposed and analyzed in

this paper. These methods, supporting by a large amount of numerical com-


putation, are fairly efficient. Comparing with Wampler�s method [12] and Li

and Bai�s method [7], the number of partitions that our methods use, reducesdramatically. That of the assembly method is even reduced to n3, which is

substantial as n increases.

Hence we are able to make the size of the problems larger in our

computations, but still limited. The reason is that we have to compute the

multi-homogeneous B�eezout number for each given partition of variables.

Wampler�s algorithm with row expansion recursion [12], which is the best

method so far to our knowledge, is used. This makes the whole computation

still too costly for large systems at present. Thus an efficient algorithm ofcomputing multi-homogeneous B�eezout number for a given partition remains

an open question.

Combinatorial geometry has come to play an important role in homotopy

methods during the last several years [2,5,6,8,11]. With some combinatorial

geometric techniques, one can get another bound, which is called the stable

mixed volume, for the number of the isolated solutions. It may be tighter than

the minimal multi-homogeneous B�eezout number. Huber and Sturmfels [5]

constructed the starting system with random lifting and convex hull techniques.Such a procedure is called polyhedral homotopy. However it is very expensive

to obtain the starting system for the current polyhedral homotopy method.

Therefore, we may obtain a partition with a satisfactory multi-homogeneous

B�eezout number within a reasonable amount of time by our methods. It may be

larger than the stable mixed volume. But the total computational cost for the

multi-homogeneous homotopy method may still be less.

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heuristic methods for computing the minimal multi-homogeneous bézout number

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