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is called the
Sylvester matrix
of
F
and
G
w.r.t.
x
. The determinant of the matrix
is called the
Sylvester resultant
or
resultant
of
F
and
G
w.r.t.
x
and is denoted
by
res(
F, G, x
)
.
For system (2), we compute the resultant of
f
s
and
f
s
w.r.t.
x
s
and denote it
by
dis(
f
s
)
(it has the leading coecient and discriminant of
f
s
as factors). Then
we compute the
successive resultant
of
dis(
f
s
)
and the triangular set
{
f
s−
1
, ..., f
1
}
.
That is, we compute
res(res(
)
,f
1
,x
1
)
and denote it by
res(dis(
f
s
);
f
s−
1
, ..., f
1
)
or simply
R
s
. Similarly, for each
i
(1
<i≤ s
)
, we compute
R
i
=res(dis(
f
i
);
f
i−
1
, ..., f
1
)and
R
1
=dis(
f
1
)
.
For each of those inequalities and inequations, we compute the successive
resultant of
g
j
(or
h
j
) w.r.t. the triangular set
[
f
1
, ..., f
s
]
and denote it by
Q
j
(resp.
Q
l
+
j
).
Definition 2.
For an SAS
T
as defined by (2), the
border polynomial
of
T
is
···
res(res(dis(
f
s
)
,f
s−
1
,x
s−
1
)
,f
s−
2
,x
s−
2
)
···
s
l
+
BP
=
R
i
Q
j
.
i
=1
j
=1
Sometimes, for brevity, we also abuse
BP
to denote the square-free part or the
set of square-free factors of
BP
.
Example 2.
For the system
S
in Example 1, the border polynomial is
BP
=
b
(
b
2)(
b
+2)(
b
2
2)(
b
4
4
b
2
+ 2)(2
b
4
2
b
2
+1)
.
−
−
−
−
From the result in [31,27], we may assume
BP
=0
.
In fact, if any factor of
BP
is
a zero polynomial, we can further decompose the system into new systems with
such a property. For a parametric SAS, its border polynomial is a polynomial in
the parameters with the following property.
Theorem 1.
Suppose
S
is a parametric SAS as defined by (2) and
BP
its border
polynomial. Then, in each connected component of the complement of
BP
=0
in
parametric space
R
d
, the number of distinct real solutions of
S
is constant.
Third,
BP
=0
decomposes the parametric space into a finite number of con-
nected region. We then choose sample points in each connected component of
the complement of
BP
=0
and compute the number of distinct real solutions of
S
at each sample point. Note that sample points can be obtained by the partial
cylindrical algebra decomposition (PCAD) algorithm [4].
Example 3.
For the system
S
in Example 1,
BP
=0
gives
±
√
2
,
2
±
√
2
.
b
=0
,
±
2
,
±
The reals are divided into ten open intervals by these points. Because
0
<b<
2
,
we only need to choose one point, for example,
2
,
1
,
2
,
1
8
, from each of the four
intervals contained in
(0
,
2)
, respectively. Then, we substitute each of the four
values for
b
in the system, and compute the number of distinct real solutions of
the system, consequently obtain the system has respectively
0
,
1
,
0
and
0
distinct
real solutions.
1
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