Download Extraneous Factors in the Dixon Resultant

Extraneous Factors in the Dixon Resultant Formulation
Deepak Kapur
Institute for Programming and Logics
Dept. of Computer Science
SUNY Albany, Albany, NY 12222
Tushar Saxena
Image Understanding Lab.
GE Corporate R & D (CMA)
Schenectady, NY 12309
[email protected]
http://www.cs.albany.edu/ kapur
[email protected]
http://www.cs.albany.edu/ saxena
Abstract
when resultants are used for elimination in a variety of applications including computer vision, robotics and kinematics,
solid and geometric modeling, geometry theorem proving,
biology, etc. This study was motivated by our attempts to
compute the exact resultant of the famous Stewart platform
problem using the generalized Dixon method.
It is proved here that for a class of polynomial systems,
the projection operator (which is a nonzero multiple of the
resultant) computed by the generalized Dixon formulation
as introduced in [11] can be completely related to the projection operator (henceforth, by a projection operator, we will
mean the one computed by the generalized Dixon method
in [11], whenever it is understood) for a simpler polynomial system (with polynomials of smaller Newton polytopes
and lower degrees). This relationship can be exploited in
many dierent ways. Firstly, the projection operator for
such smaller polynomial systems can be computed much
faster using less memory space, than similar computations
for the larger systems. Secondly, extraneous factors in the
projection operator of a given polynomial system can also
be completely determined from the extraneous factors in
the projection operator of the related smaller polynomial
system. As a corollary, if the generalized Dixon formulation
does not generate extraneous factors for the smaller system,
then it is guaranteed not to do so for the larger one either.
It therefore becomes useful to identify polynomial systems
for which the generalized Dixon formulation computes the
exact resultant.
In this direction, we identify families of systems, both
unmixed and mixed, for which the projection operator contains no extraneous factors. As an example, we discuss the
Stewart Platform problem [12]. We show how the generalized Dixon method can be used to compute the exact resultant of degree 40 of the Stewart Platform problem. This
is accomplished by establishing the existence of a variable
ordering under which the projection operator is of degree
40 and hence, has no extraneous factor [15, 12, 13], and is
exactly its resultant. To the best of our knowledge, it is the
rst time that a general resultant method has been directly
applied to this problem without producing any extraneous
factor.
The results reported in this paper extend our earlier results about the generalized Dixon formulation, a method for
simultaneously eliminating many variables for a large class
of polynomial systems and computing a projection operator
from which the resultant can be extracted [11]. This method
has been experimentally found to be superior in performance
Elimination methods based on generalizations of the Dixon's
resultant formulation have been demonstrated to be ecient
for simultaneously eliminating many variables from polynomials. One of these methods, presented by the authors earlier, was even shown to exploit the sparse structure of a
polynomial system as determined by its Newton polytope.
This paper analyzes the extraneous factors in the projection
operators computed by that method. It is shown that the
projection operator of a polynomial system can be related to
the projection operator of another system consisting of polynomials with smaller Newton polytopes and lower degrees,
thus making resultant computation more ecient. If a larger
polyomial system can be obtained from another smaller one
by replacing variables by their powers, then the projection
operator of the larger system is proved to be a power of the
projection operator of the smaller one. This shows that the
set of extraneous factors in the two projection operators are
the same, though with dierent multiplicities. This implies
that identifying the projection operator (and its extraneous
factors) of a given polynomial system determines the projection operator (and its extraneous factors) of a whole family
of polynomial systems, viz., the one generated from this system by replacing variables by their powers. Such results determining a priori extraneous factors, or the absence thereof,
by examining smaller polynomial systems whose extraneous
factors may be easily identiable, are likely to help in many
application domains. It is also shown that the projection
operator, as computed by this generalized Dixon method,
of many specic polynomial systems (both mixed as well as
unmixed), is exactly their resultant. Finally, the generalized Dixon method is used to compute the exact resultant
of degree 40 for the Stewart Platform problem.
1 Introduction
This paper analyzes extraneous factors arising in resultant
computations of polynomial systems, a key problem faced
Partially Supported by the NSF grant no.
CCR-9622860
1
on a wide variety of examples, in comparison with other
elimination methods including Macaulay resultants, sparse
resultants [3, 17, 7], the characteristic set construction [5],
and the Grobner basis construction [1, 2]. The method takes
less time, less space, and seems to generate fewer extraneous factors [10] (except in the case of the Grobner basis
method which gives the exact resultant). Recently, we have
also been able to show that for the unmixed case, the Dixon
formulation, in fact, implicitly exploits the sparse structure
of the polynomial system, i.e., its computational complexity
is governed by the Newton polytope of the unmixed system, not by the Bezout bound as is the case for Macaulay
resultants [9].
The problem of extraneous factors is not peculiar to elimination methods based on the generalized Dixon formulation. In fact, none of the resultant based methods { the
Macaulay formulation, Dixon formulation or the sparse resultant formulation, compute the exact resultant of arbitrary nonhomogeneous polynomial systems. Instead, these
methods compute various multiples of the resultant, known
as projection operators, which may contain extraneous factors besides the resultant.
Since the information about the solutions of a polynomial
system is completely contained in its resultant, the extraneous factors do not oer any additional information. Instead,
they make it more dicult to identify the resultant in a projection operator, because each of its factors must be checked
to determine if it is a part of the resultant or not. This check
can be resource-consuming. The presence of extraneous factors can also make projection operator computation impractical, since they increase the total degree of the projection
operator considerably, and the computational complexity of
the projection operator using the generalized Dixon method
(as well as other resultant formulations) is determined by its
degree.
Our overall objective is to obtain results and develop
heuristics which aid in identifying and even eliminating extraneous factors. In this paper, we prove that if the variables
of a polynomial system are scaled (ie. they are replaced by
their powers), then the projection operator of the resulting
system is exactly that of the original, raised to the product
of the powers by which the individual variables were scaled.
This shows the generalized Dixon formulation does not introduce any new extraneous factors under variable scaling.
As a consequence, identication of the resultant and extraneous factors in the projection operator of a given system
precisely determines the resultant and extraneous factors in
any system which can be obtained from this one by variable
scaling, and they need not be recomputed. Alternatively,
this also suggests that before computing projection operators polynomial systems should be scaled down as much as
possible. We also identify polynomial systems for which the
projection operator is the resultant itself (i.e., it does not
contain any extraneous factors). The systems discussed in
this paper include the Sum of Square roots problem, the Binary Representation problem, their generalizations and the
Stewart Platform problems.
The results presented in this paper about the presence or
absence of extraneous factors in projection operators computed by multivariate resultant formulations are the rst
such, to the best of our knowledge. In [4, 8], authors identied the precise relationship between the resultant of a
system of homogeneous polynomials and another obtained
by replacing the variables by polynomials. For the type
of transformation discussed in our paper, the relationship
identied by Hong between the resultants of the two systems turns out to be exactly the same as the relationship
identied by our results between their projection operators.
This shows that the generalized Dixon method is optimal
under such transformations because if the projection operator of the smaller system is exactly the resultant, then such
a transformation does not generate any extraneous factors
when the method is applied to the larger scaled system.
In Section 2, the generalized Dixon method is reviewed.
The resultant of a system of a nonhomogeneous polynomials is dened. The denitions of Cancellation matrix, Dixon
polynomial, Dixon matrix and projection operator are reproduced from [11, 10].
In Section 3, the projection operator computed by the
generalized Dixon method for a given polynomial system is
related to the projection operator of another polynomial system obtained by replacing variables in the original system
by their powers (of degrees, say d ). We call this process
variable scaling. We show in this section that the projection
operator of the larger system is a power of the projection
operator of the smaller system, where this power is the product of the degrees, d . If the extraneous factors (or the lack
thereof) in the projection operator of any specic system
have been identied, then using our results, the extraneous
factors (or the lack thereof) have been eectively identied
in the projection operators of a whole family of polynomial
systems which is obtained from this system under such variable scalings. It thus becomes essential to identify extraneous factors in the projection operators of specic polynomial
systems. In the next few sections, we do precisely that.
Section 4 discusses a subclass of unmixed polynomial
systems for which the projection operator is precisely the
resultant, i.e., it does not have any extraneous factors. Examples of those unmixed polynomial systems are also given
for which the projection operator computed using the generalized Dixon method includes extraneous factors no matter
what variable ordering is used.
The case of mixed polynomial systems is a little more
dicult. Section 5 discusses some specic examples of mixed
systems whose resultant can be exactly computed using the
generalized Dixon method. This subclass includes the sum
of square roots problem proposed to us by Chee Yap as
well as binary integer representation problem (which is a
generalization of the knapsack problem) suggested to us by
Laureano Gonzalez Vega.
Section 5 also includes our experience in using the generalized Dixon method on the Stewart platform problem which
motivated us to look at the problem of extraneous factors
and hence, this paper. For the Stewart Platform problem,
we show the existence of a variable ordering under which the
projection operator computed using the generalized Dixon
formulation is of degree 40 and hence, has no extraneous
factor [15, 12, 13], and is exactly its resultant. To the best
of our knowledge, it is the rst time that a general resultant
method has been directly applied to this problem without
producing any extraneous factor.
i
i
2 The Dixon Formulation
Let X = fx ; : : : ; x g be a set of n variables and P =
fp ; ; p g a set of n +1 polynomials in X . By a polyno1
1
n+1
n
mial system we will mean this set of n + 1 polynomials in n,
unless otherwise stated. The coecients of the polynomials
in P are assumed to be polynomials from Q[A], where A is a
2
set of indeterminates we call parameters. Note that the coecients of polynomials in P could be numbers themselves.
Newton Polytope of a polynomial p in variables X is
the convex hull of the set of exponents (treated as points in
R ) of all terms in p. A polynomial system is unmixed if
all its polynomials have the same Newton polytope [6, 9]. A
polynomial system is generic if each coecient in the system
is an independent indeterminate.
Let I be the ideal generated by P in Q[A;X ]. All polynomials in the ideal J = I \ Q[A] are known as the projection
operators of P with respect to X . Clearly, projection operators of P vanish at all those specializations of A from
, of Q, for which P has a common
the algebraic closure, Q
. Note that 0 is trivially always a projection
solution in Q
operator of all P . The resultant of P is dened to be the
unique (upto a scalar multiple) generator of the ideal J .
The resultant of P is 0 if and only if P has no nonzero projection operator. A nonzero resultant of P must divide all
its projection operators. Finally any factor of a projection
operator of P which is not in its resultant is known as an
extraneous factor.
Most ecient resultant formulations compute a projection operator, not the resultant. Thus they also tend to
produce extraneous factors, which are undesirable. Our objective in this paper is to derive some properties of these
extraneous factors which will enable their identication, or
even their elimination while the projection operator is being
computed using the generalized Dixon formulation.
First we review the generalized Dixon formulation for
computing the projection operator of P . X is a set of n new
variables fx1 ; : : : ; x g. The Cancellation matrix C of P
is the following (n + 1) (n + 1) matrix:
2 p (x ; x ; : : : ; x ) p
3
1
1
2
+1 (x1 ; x2 ; : : : ; x )
x1 ; x2 ; : : : ; x ) p +1 (x1 ; x2 ; : : : ; x ) 7
6 p1 (
C = 66 p1 (x1 ; x2.; : : : ; x ) p +1 (x1 ; x.2 ; : : : ; x ) 77 ;
4
5
..
..
p1 (x1 ; x2 ; : : : ; x ) p +1 (x1 ; x2 ; : : : ; x )
operator; this is so because all other entries in the Dixon
matrix of the transformed polynomial system are identically
zero. However, for simplicity of proofs and readability, we
will assume in the remaining paper that the Dixon matrix
is square and non-singular.
Some obvious facts follow: The entries of the Cancellation matrix are polynomials in X [X . The Dixon polynomial
is a polynomial in X [ X . The entries of the Dixon matrix
are polynomials in the coecients of P , and the projection
operator is also a polynomial in the coecients of P .
n
3 Projection Operator under Variable Scaling
In this section, we will analyze the change in the projection
operator when variables in a polynomial system are replaced
by their powers. We call this process variable scaling, since
such transformations have the eect of scaling the Newton
polytope of a polynomial along each variable axis.
Variable scaling is expressed as a Transformation, T , of
variables parametrized by an n-tuple of integers hf1 ; : : : ; f i.
This transformation is eected by replacing all occurences
of x and x in an expression by x i and x j respectively.
If p is a polynomial in X [ X , T (p) denotes a polynomial
in X [ X , obtained by applying T to p, as above. Given
a polynomial system P , T (P ) is obtained by replcing each
polynomial q 2 P by T (q). Similarily, if M is a matrix
whose entries are polynomials in X [ X , then T (M ) is the
matrix obtained by replacing each entry M [i; j ] of M by
T (M [i; j ]).
Following results relate the projection operators of polynomial systems T (P ) and P .
Lemma 3.1 Let be the Dixon polynomial of an arbitrary
polynomial system P . Let T be a transformation of type
hf1 ; : : : ; f i. Then the Dixon polynomial of T (P ) is
n
n
n
i
P
P
n
n
n
n
n
n
n
n
n
n
n
k
j
k
i
=
i
i
fi
!
T ( ) :
i
P
i
Proof : Given a transformation T = hf1 ; : : : ; f i, it follows
from the denitions in the previous section that T (C ) is
the cancellation matrix of T (P ), where C is the cancellation matrix of P . Also,
T ( ) = Q jT? (Ci )j i :
x ? x
=1
P
P
n
P
P
P
= Q jC(x j? x ) :
=1
P
P
T (P )
? x
x ? x
n
Y
xfi
i=1
i
n
i
j
n
i
i
f
i
n
n
j
i
f
P
where p (x1 ; ; x ; x +1 ; : : : ; x ) stands for uniformly replacing x by x for all 1 j k n in p .
) is a zero of jC j,
Q Since for all 1 i n, (x ? x
(
x
?
x
)
divides
j
C
j
.
The
Dixon
polynomial, =1
of P is the following polynomial in X [ X :
i
j
n
i
i
P
i
P
Let V be a column vector of all monomials in X which
appear in , when viewed as a polynomial in X . Similarily,
let W be a row vector of all monomials in X which appear in
, when viewed as a polynomial in X . The Dixon matrix,
D of P is the matrix for which = V D W .
If the Dixon matrix is square and non-singular, then its
determinant is a projection operator of P . The vanishing
of the projection operator is a necessary condition for P to
have a common solution in X .
One note of caution here: It is possible for the Dixon
matrix to be rectangular, or even if it is square, to be singular. In that case, the rank submatrix contruction, RSC,
described in [11] can be used to extract identically non-zero
projection operators from D .
Theorem 3.2 of this paper is applicable even when D
is rectangular, and RSC is used to extract the projection
f
i
i
i
P
P
P
f
The lemma follows from the Dixon polynomial of T (P ):
( ) = Q jT ((xC ?)j x )
=1
!
i
Y x i ?x
jT? (C )j :
=
Q
x
?
x
x i ? x i
=1
=1
P
P
n
T P
P
n
i
i
n
f
i
i
i
f
P
i
i
i
n
f
f
i
i
i
2
Theorem 3.2 Let T be a transformation of type hf ; : : : ;
f i. The
projection operator of T (P ) is exactly the
?Q
1
n
P
n
i=1
P
3
f
i
th
power of the projection operator of P .
Proof :
Example: Consider the polynomial system P = fp ; p ;
p g in variables X = fx ; x g, where:
p = ?4x x + ax + 15x x + 17;
p = 10x x + bx ? 19x x + 4;
p = ?7x + 2x x + c;
By the construction in the Dixon Formulation,
1
= V D W , where V and W are column and row vectors
whose entries are power products in X and X respectively,
and D is the Dixon matrix of P . By Lemma 3.1, the Dixon
polynomial of T (P ) is
P
3
P
T (P )
=
x ? x
x ? x
=1
fi
i
x ? x
x ? x
=1
n
Y
fi
i
i
i
Qn
?
i
fi
2
!
3
T ( )
i
i
i
=
fi
!
T (V ) D T (W ) :
i
P
Qn
The product =1 x i ? x i = (x ? x ) has =1 f monomials in X [ X , and a little algebraic manipulation of this
Dixon polynomial of T (P ) leads to the fact that the Dixon
matrix of T (P ) is the following matrix:
2 D
0 0 0 3
0 0 7
6 0 D
D ( ) = 664 0. 0. D. 0. 775 :
..
..
.. ..
0 0 0 D
i
f
f
i
i
i
i
i
i
P
P
k
P
f
So, we nd that scaling down a polynomial system, as
determined by our results and described in our technique,
indeed helps reduce the computational complexity of resultant computation without losing any information about the
resultant.
n
i
th
i
3.1 Optimizing Projection Operator Computation
4 Extraneous Factors in Unmixed Systems
These results give us the following technique to reduce both,
the cost of computing a projection operator, and also the
multiplicity of extraneous factors in it. Given a polynomial
system P , rst compute c , the greatest common divisor of
all powers of x occurring in the non-zero terms of P . Then,
for all 1 i n, divide each monomial in P which contains
x i , for some k > 0, by x ( i ?1) . Finally, compute the
projection operator of the resulting smaller system.
This technique basically nds the smallest system that
P can scale down to, and then, works with that smaller
system. Since such a procedure reduces the volume of the
Newton polytope of the input polynomials, it requires less
computational resources than directly computing the projection operator of the larger system [9]. This procedure
also reduces the total degree of the projection operator by
reducing the power of each extraneous factor and the resultant in the projection operator. Let us illustrate this with
an example.
It is important to identify polynomial systems whose projection operators do not have extraneous factors. Such systems, in conjunction with the results of the previous section, can then be used to identify a family of polynomial
sets (using variable transformations), whose projection operators do not have extraneous factors. We will illustrate
this point further later (specically, Corollary 4.3). In this
section, we present some results about extraneous factors in
the projection operators of unmixed systems.
It should be pointed out that the projection operator
generated by our method depends upon the order in which
the variables in X are replaced by their counterparts in X ,
in the cancellation matrix. Dierent variable orders produce
dierent cancellation matrices, thus dierent Dixon polynomials, Dixon matrices, and nally dierent projection operators. Some of the following results will be true for all
variable orderings, and we will state them so.
i
i
kc
k c
i
i
k
q1 = ?4x21 x2 + ax22 + 15x1 x2 + 17
q2 = 10x21 x2 + bx1 ? 19x1 x32 + 4
q3 = ?7x32 + 2x31 x22 + c:
Indeed, our program only took 0:2 seconds to construct the
Dixon matrix of this smaller system, whose size was only
13 13. It took a total time of only 65 seconds to compute
its projection operator, whose total degree in the parameters was found to be only 23 = 460=20. In the projection
operator
of the smaller system, there was an extraneous factor
b2 , which tells us that there will be an extraneous factor
? 2 20
b = b40 in the projection operator of the larger system
P too.
2
P
P
n
k
i
This block diagonal matrix has =1 f occurences of D
along the diagonal. The
Qn projection operator of T (P ) is its
determinant, ie. jD j i=1 i :
2
A polynomial system P is said to scales down to
the system Q if there exists a transformation T such that
T (Q) = P . Above results are signicant because they immediately enable one to infer the following corollaries about
extraneous factors.
Corollary 3.3 If P scales down to Q, the projection operators of P and Q have the same set of extraneous factors,
though their multiplicities may be dierent.
Corollary 3.4 If P scales down to Q via a transformation
T of type hf1 ; : : : ; f i, and if the projection operator of Q is
its resultant, i.e., does not have any?Q
extraneous
factors, then
f
power of its
the projection operator of P is the
=1
resultant, and does not have any extraneous factors.
i
5 4
1 2
5 12
1 2
k
Qn
i
8
2
5
1
15 8
1
2
th
P
T P
2
2
from which x1 and x2 are to be eliminated to compute resultant, a polynomial in a; b and c. We used a MAPLE
implementation of our method based on the Dixon formulation [10] to compute the projection operator. The size of
the Dixon matrix of P was 260 260 and it took 14:5 seconds to construct it. The total time taken to compute the
projection operator of P was 3008 seconds. The degree of
the projection operator in parameters a; b and c was 460.
However, using the technique based on our results from
previous section, we nd that the GCD of the powers of x1
in P is 5, and the GCD of the powers of x2 is 4. Therefore,
according to our results, the projection operator of P is the
5 4 = 20 power of the projection operator of the smaller
system5 obtained
by dividing all monomials in P which4 con4
tain
x
by
x
by
1
1 , and all monomials which contain x2
x32 . This smaller system contains polynomials fq1 ; q2 ; q3 g,
where p = T (q1 ) and T = h5; 4i. Therefore:
P
i
10 4
1
2
10 4
1
2
12
2
1
P
n
Y
1
4
Corollary 4.3 If a polynomial system is completely hollow,
First let us present a fact about extraneous factors, which
may be viewed as a somewhat negative result.
There exist generic, unmixed systems such that
under all variable orderings, their projection operator contains extraneous factors.
generic, unmixed, and if its Newton polytope is the same as
some generic n-degree polynomial system, then its projection
operator is exactly its resultant.
Example: Consider the following system:
Example: Consider the generic unmixed system P =
fp ; p ; p ; p g in variables X = fx ; x ; x g, where:
1
2
3
4
1
2
p1 = a1 x31 x22 + a2 x31 + a3 x22 + a4
p2 = b1 x31 x22 + b2 x31 + b3 x22 + b4
p3 = c1 x31 x22 + c2 x31 + c3 x22 + c4 :
3
p1
p2
p3
p4
= a1 x23 x22 + a2 x22 x1 + a3 x3 x2 + a4 x23 + a5 ;
= b1 x23 x22 + b2 x22 x1 + b3 x3 x2 + b4 x23 + b5 ;
= c1 x23 x22 + c2 x22 x1 + c3 x3 x2 + c4x23 + c5 ;
= d1 x23 x22 + d2 x22 x1 + d3 x3 x2 + d4 x23 + d5 :
No matter which of the 6 variable orderings is used, the
Dixon matrix of P is 5 5 with full rank of 5 and the projection operator of P , as computed by the generalized Dixon
formulation, always has an extraneous factor which equals
a3 b2 c6 d1 ? a6 b2 d1 c3 ? a6 b1 c2 d3 + a6 b1 d2 c3
+ a3 b6 d2 c1 ? a6 b3 d2 c1 + a2 b6 d1 c3 ? a3 b2 d6 c1
? a3 b6 c2 d1 ? a3 b1 d2 c6 ? a2 b1 d6 c3 + a3 b1 c2 d6
+ a6 b2 c1 d3 + a6 b3 c2 d1 + a1 b6 c2 d3 ? a1 b6 d2 c3
? a2 b6 c1 d3 ? a1 b2 c6 d3 ? a2 b3 c6 d1 + a1 b2 d6 c3
+ a2 b3 d6 c1 + a1 b3 d2 c6 + a2 b1 c6 d3 ? a1 b3 c2 d6 :
This is a completely hollow, generic, unmixed system whose
Newton polytope is the same as that of a system of n-degree
(3; 2). Indeed, its projection operator, as computed by the
generalized Dixon formulation, has no extraneous factors, in
accordance with Corollary 4.3.
2
Notice that most of our results have required all coecients to be generic, i.e., independent indeterminates. Rojas
[14] has recently obtained results characterizing precisely the
coecients in a polynomial system which need to be chosen
generically for the number of roots of a given system to be
exactly equal to the derived upper bounds. These results
may also be useful in strengthening our results by relaxing
the requirement that all coecients be generic.
5 Examples of Good Mixed Systems
In this section, we identify some mixed systems whose projection operator is exactly their resultant. A system is called
mixed if all its polynomials have dierent Newton polytopes. Again, by replacing individual variables by their
powers, these systems can be viewed as generating a family
whose projection operators do not contain any extraneous
factors. However, the systems discussed in this section are
important in their own right, as they appear in dierent
applications. One of the surprising result we will present
is to identify a variable ordering on the Stewart platform
problem, for which its projection operator, as computed
by the generalized Dixon formulation, is exactly its resultant. Stewart platform is a benchmark elimination problem
in kinematics, and to the best of our knowledge, our result
is the rst time that a general resultant method has been
applied to this problem without generating any extraneous
factors.
2
This implies that it is not always possible to nd at least
one variable ordering under which the projection operator is
exactly the resultant { even if the polynomial system under
consideration is a generic unmixed system.
A system of n + 1 nonhomogeneous polynomials p1 ; : : : ;
p +1 in n variables, x1 ; : : : ; x is called generic n-degree
if there exist nonnegative integers d1 ; : : : ; d such that
n
n
n
p =
d1
X
j
i1 =0
dn
X
a
j;i1 ;::: ;in
x11 x for 1 j n + 1;
i
in
n
in =0
where each coecient a 1 n is a distinct indeterminate.
(d1 ; : : : ; d ) is known as their n-degree. An arbitrary ndegree system is simply one whose Newton polytope equals
that of some generic n-degree system. The following theorem is proved in [16]:
Theorem 4.1 The projection operator of a generic ndegree system is exactly its resultant, that is, it has no extraneous factors, under all variable orderings.
Together with Theorem 3.2, we have the corollary:
Corollary 4.2 If a polynomial system P can be scaled down
to a generic n-degree system, then the projection operator of
P does not have any extraneous factors.
It directly follows from Theorem 4.1 that the projection
operator of a generic system of n-degree (1; : : : ; 1) is its resultant. Since any n-degree system which is completely hollow
(dened to be a polynomial system whose polynomials contain the minimum number of terms required to maintain its
Newton polytope) can be scaled down to a generic system
of n-degree (1; : : : ; 1), it follows that:
j;i ;::: ;i
n
5.1 Sum of Square Roots and Related Problems
Given a set of n positive integers, s1 ; : : : ; s , the objective
here is to compute a polynomial, all of whose roots are exactly the sums of all possible square roots of the s . This
problem was communicated to us by Chee Yap, and can
easily be posed as a resultant computation problem.
Let = fA1 ; A2 ; : : : ; A +1 g be the following system
of n + 1 polynomials:
A = x2 ? s
1 i n;
n
i
n
n
i
A
n+1
i
= y?
i
n
X
x:
i
i=1
The resultant, R , of with respect to variables x1 ; : : : ; x
is a polynomial in the parameters y; s1 ; : : : ; s : It is easily
n
n
n
n
5
seen that the degree of the resultant in y is 2 . R can be
dened recursively as
R = subs (y = y + pp
s ; R ?1 )
subs (y = y ? s ; R ?1 ) ;
where subs (y = z;p) is the polynomial obtained by replacing all occurences of y by z in p. We now dene a matrix
whose determinant is this resultant; such matrices are called
resultant matrices.
n
n
n
n
n
n
=
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
=
n
x2 ? s
x2 ? s
n
n
n
..
.
x ? s
2
n
n
n
x ? s
x ? s y ? x ?
..
.
2
1
2
1
1
..
.
1
1
Qx ? s
y?
(x ? x )
2
1
n
i=1
1
i
..
P.
i=2
n
i=1
x
i
x
i
?1
n
n
n+1
i
i
i
= u0 +
i
n
Pn
xi
y? P
i=1
n
n
n
n
X
i
n
i
ux;
i
i
n
n
x A ?1 . The base case ie, 1 = X1 D1 A1 is easy. Let
A ?1
?1 = X ?1 D ?1 A ?1 .
n
A
n
n
n
n
n
n
x A ?1
the Dixon matrix is of size 2 2 and is optimal. And,
the projection operator of this system as computed by the
Dixon formulation has no extraneous factors.
Another example of a problem (suggested to us by Laureano Vega) in this class is that of determining the binary representation of an integer. If hx1 ; : : : ; x i (where x 2 f0; 1g)
is the binary representation of an integer y, then it must be
a solution of the polynomial system:
A = x2 ? x
1 i n;
n
n
n
i=1
n
n
n
n
n
A
n
n
n
n
i
It can be shown that the Dixon matrix for this problem
equals D .
Theorem 5.3 D is the Dixon Matrix of .
Proof : By Induction, we prove that the Dixon polynomial, = X D A , where we recursively dene
X1 =
x
1
[1; x1 ] ; X = [X ?1 ; x X ?1 ] and A1 = 1 ; A =
n
n
i
n
n
n
n
i
n
n
n
n
The above analysis concerning the optimality of the Dixon
resultant matrix generalizes even when A , for 1 i n,
are arbitrary second degree polynomials in x and A +1 is
an arbitrary linear polynomial. I.e., for a system
A = q x2 + r x + s
1 i n;
n
n
n
n
n
5.1.2 Binary Representation Problem
n
n
n
i
n
n
n
i
n
n
n
n
(x + x )
n
n
n
?1
Y
n
n
n
n
n
n
n
n
D ?1
I2 ?1
= j(D ?1 + ps I2 ?1 ) (D ?1 ? ps I2 ?1 )j
= subs (y = y + ps ; jD ?1 j)
subs (y = y ?pps ; jD ?1 j)
= subs (y = y + s ; R ?1 )
subs (y = y ? ps ; R ?1 )
= R:
The degree of R in y is 2 . Since y is the constant coecient of polynomial A +1 , all resultant matrices whose
entries are coecients of polynomials in (or constants)
must at least be of size 2 2 . D is exactly of this size,
hence it is the smallest.
2
n
n
2
sn I2n?1 = Dn2 ?1 ? sn I n?1 2
Dn?1 n
n
i
i
From Theorems 5.2 and 5.3, it follows that
Corollary 5.4 (i) The Dixon matrix of is the optimal
resultant matrix with the coecients of terms in as its
entries and (ii) The projection operator of computed by
the Dixon method is its exact resultant, and has no extraneous factors.
In contrast, the size of the sparse resultant
of is
? matrix
2 + n 2 ?1 , and of Macaulay matrix is 2 +1 . Since these
matrices are not of optimal size, it suggests that there may
be extraneous factors in the projection operators computed
by these methods.
n
n
n
n
n
n
n
n
n
n
n
n
n
= [X ?1 ; x X ?1 ] IDn??11 s DI2?n?1 1
2
= X DA
entries are purely the coecients of polynomials in .
Proof : First note that all entries on the major diagonal of
D ?1 are y. The proof is by induction on n. The base case,
jD1 j = R1 is easy. Assume jD ?1 j = R ?1 .
n
n
n
n
n
= (x + x ) X ?1 D ?1 A ?1
+ (s + x x ) X ?1 I2n?1 A ?1
n
k
n
n
n
i=1
n
n
n
n
n
n
n
n
n
n
D0 = [y]
D = ID2 ??11 s DI2??1 1 ; 8n 1;
where I be the k k identity matrix.
Theorem 5.2 (i) D is a resultant matrix of , ie., R =
jD j and (ii) D is the smallest resultant matrix of whose
jD j =
0
x1 + x1 ?1 0
0
?1 ..
..
..
..
.
.
.
.
x ?1+ x ?1 0
?1 0
0
?1
P ?1 ?s ? x x x2 ?1? s ?1 x21 ?s1 y ? =1
x
0
0
..
.
0
x + x
= (x + x ) ?1 + (s + x x )
5.1.1 The Resultant Matrix and the Dixon Matrix
Denition 5.1 For n 0, D is a 2 2 matrix dened
recursively as:
A
n+1
i
= y?
i
i
n
X
2 ?1 x :
i
i
i=1
Therefore, the resultant of this system, obtained by eliminating all the x is a polynomial in y, whose roots are all
integers whose binary representation take no more than n
bits.
i
i
6
5.2 The Stewart Platform Problem
quite sensitive to the ordering on the variables. The sensitivity of the performance of the method to the ordering is
quite evident in our theoretical results [9]: the complexity of
the method is determined by the volume of the Minkowskii
sum of successive projections of the Newton polytope. The
order in which successive projections are done can change
the resulting volume.
Emiris reported in his thesis [6] that the sparse resultant
matrix of the above problem formulation is 405 405. We
could not nd in his thesis the time it took his program to
construct this matrix. Moreover, he reported (see section 5.3
in [6]) that the univariate matrix polynomial corresponding
to this matrix is identically singular, therefore it seems that
the method in [6] may be unable to nd the roots, or the
resultant of this system using the sparse resultant formulation. Due to lack of memory space, we have not been able
to construct a sparse resultant matrix for this system on a
SPARC station 10 using Emiris' program.
To our knowledge, our results are the rst successful attempt to compute the resultant for the Stewart platform
problem using multivariate elimination methods in which
no a priori processing or ad hoc techniques are employed.
The Stewart platform problem is a well-known benchmark
from robotics and kinematics which can be formulated as a
multivariate elimination problem [12, 13, 6]. We show how
the generalized Dixon method can be used to compute exactly the polynomial giving all the roots. This problem is
also quite interesting in illustrating another aspect of elimination methods, namely even though in a multivariate resultant formulation, variables are eliminated simultaneously,
they implicitly assume an ordering on variables which signicantly aects their computational performance as well as
extraneous factors generated in the projection operators. In
particular, we show that, for certain orderings, the generalized Dixon method computes the result quite fast. However,
such orderings result in a projection operator which contains
numerous extraneous factors. On the other hand, another
ordering takes longer, but gives the exact resultant, without
any extraneous factors! Therefore, using the results of our
paper, if a system is obtained from the Stewart platform system by variable scaling, then its projection operator will not
contain any extraneous factor, as long as the latter ordering
is used.
The quaternion formulation of the Stewart platform
presented below is by Emiris [6]. It contains 7 polynomials in 7 variables, 6 of which are to be eliminated.
Let x = [x0 ; x1 ; x2 ; x3 ] and q = [1; q1 ; q2 ; q3 ] be two unknown quaternions, which are to be determined. Let q =
[1; ?q1 ; ?q2 ; ?q3 ]. Let a and b , for i = 2; : : : ; 6, be ten
known quaternions, and let , for i = 1; : : : ; 6, be six known
scalars. The 7 polynomials generated for solving the Stewart
platform problem are:
i
References
[1] Becker T., Weispfenning V., in cooperation with Kredel H., Grobner Bases, A Computational Approach
to Commutative Algebra, Springer-Verlag, New York,
1993.
[2] Buchberger B., Grobner bases: An Algorithmic Method
in Polynomial Ideal Theory, Multidimensional Systems
Theory, N.K. Bose, ed., D. Reidel Publ. Co., 1985.
[3] Canny J. and Emiris I., An Ecient Algorithm for
the Sparse Resultant, Proc. of AAECC 93, LNCS 263,
Springer-Verlag, May 1993, pp. 89{104.
[4] Cheng C.C., McKay J. and Wang S.S., A chain rule
for multivariate resultants, Proc. Amer. Math. Soc. 123
(1995), pp 1037{1047.
[5] Chou S.C, Mechanical Geometry Theorem Proving,
Reidel, Dord recht, 1988.
[6] Emiris I., Sparse Elimination and Applications in Kinematics, Doctoral Thesis, Computer Science Division,
University of Calif., Berkeley, 1994.
[7] Gelfand I.M., Kapranov M.M. and Zelevinsky A.V.,
Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, 1994.
[8] Hong Hoon, Multivariate Resultants Under Composition, Technical Report, RISC Linz, Austria, 1996.
[9] Kapur D and Saxena T., Sparsity Considerations in the
Dixon Resultant Formulation, Proc. ACM Symposium
on Theory of Computing, Philadelphia, May 1996.
[10] Kapur D and Saxena T., Comparison of Various Multivariate Resultant Formulations, Proc. ACM International Symposium on Symbolic and Algebraic Computation, Montreal, July 1995.
[11] Kapur D., Saxena T. and Yang L., Algebraic and Geometric Reasoning using Dixon Resultants, Proc. ACM
International Symposium on Symbolic and Algebraic
Computation, Oxford, England, July 1994.
i
i
f1 = x x ? 1 q q;
f = b (xq) ? a (qx) ? (qb q ) a ? q q;
i = 2; : : : ; 6;
f7 = x q :
Out of the 7 variables, x0 ; x1 ; x2 ; x3 ; q1 ; q2 ; q3 , any six are to
T
i
T
i
T
T
i
i
T
i
i
T
T
be eliminated to compute the resultant as a polynomial in
the seventh. It is known (see [15, 12, 13]) that this system
has exactly 40 complex roots, therefore the resultant is a
polynomial of degree 40.
Using our implementation of the generalized Dixon
method running on a sparc 10 station with 64MB memory
(see [16]), we successfully computed the projection operator from the above formulation. Our experiments show that
if we eliminate x0 ; : : : ; x3 ; q2 ; q3 from this system, then its
Dixon matrix is 75 107. The method takes 94:65 seconds
to construct it. Rank of the Dixon matrix is 56, and it
takes 71:06 seconds to interpolate a projection operator of
degree 103 in q1 , implying that there are extraneous factors of degree 63 ( = 103 ? 40). When the ordering is
changed to x1 ; : : : ; x3 ; q1 ; : : : ; q3 , the size of the Dixon matrix
increased to 165 219. Even the rank of this matrix, which
is 116, is larger than the dimensions of the Dixon matrix
obtained from the previous ordering. Since this Dixon matrix is larger, it takes 273:5 seconds to construct it. It also
takes 256:79 seconds to interpolate a projection operator.
However, this projection operator, which is a polynomial in
x0 , only has degree 40, which implies that it does not have
any extraneous factor.
This shows that the generation of extraneous factors as
well as the computational performance of the method are
7
[12] Lazard D., Generalized Stewart Platform: How to compute with rigid motions?, Proc. IMACS-SC, 1993.
[13] Mourrain B., The 40 \Generic" Positions of a Parallel
Robot, Proc. ACM International Symposium on Symbolic and Algebraic Computation, Kiev, July 1993.
[14] Rojas J.M., Toric Intersection Theory for Aine
Root Counting, Preprint, Department of Mathematics, MIT, 1996. Also available online at http://wwwmath.mit.edu/rojas.
[15] Ronga F. and Vust T., Stewart Platforms without Computer, Preprint, Department of Mathematics, University of Geneva, 1992.
[16] Saxena T., Ecient Variable Elimination using Resultants, Doctoral Thesis, Department of Computer Science, State University of New York at Albany, 1996.
[17] Sturmfels B., Sparse Elimination Theory, Proc. Computat. Algebraic Geom. and Commut. Algebra, D. Eisenbud and L. Robbiano, eds., Cortona,Italy, Cambridge
Univ. Press, June 1991.
8