Citation for published version:
Chen, C, Davenport, JH, Moreno Maza, M, Xia, B & Xiao, R 2011, Computing with semi-algebraic sets
represented by triangular decomposition. in ISSAC '11 Proceedings of the 36th International Symposium on
Symbolic and Algebraic Computation. Proceedings of the International Symposium on Symbolic and Algebraic
Computation, ISSAC, Association for Computing Machinery, New York, pp. 75-82, 36th International Symposium
on Symbolic and Algebraic Computation, ISSAC 2011, June 8, 2011 - June 11, 2011, San Jose, CA, USA
United States, 1/01/11. https://doi.org/10.1145/1993886.1993903
DOI:
10.1145/1993886.1993903
Publication date:
2011
Document Version
Peer reviewed version
Link to publication
© ACM, 2011. This is the author's version of the work. It is posted here by permission of ACM for your personal
use. Not for redistribution. The definitive version was published in ISSAC 2011 - Proceedings of the 36th
International Symposium on Symbolic and Algebraic Computation. New York: Association for Computing
Machinery, pp.75-82, http://dx.doi.org/10.1145/1993886.1993903
University of Bath
Alternative formats
If you require this document in an alternative format, please contact:
openaccess@bath.ac.uk
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners
and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
Take down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Download date: 10. Aug. 2022
Computing with Semi-Algebraic Sets Represented by
Triangular Decomposition
Changbo Chen
James H. Davenport
Marc Moreno Maza
University of Western Ontario
University of Bath
University of Western Ontario
cchen252@csd.uwo.ca
J.H.Davenport@bath.ac.uk
moreno@csd.uwo.ca
Bican Xia
Peking University
xbc@math.pku.edu.cn
ABSTRACT
This article is a continuation of our earlier work [3], which
introduced triangular decompositions of semi-algebraic sys-
tems and algorithms for computing them. Our new contri-
butions include theoretical results based on which we obtain
practical improvements for these decomp osition algorithms.
We exhibit new results on t he theory of border polynomials
of parametric semi-algebraic systems: in particular a geo-
metric characterization of its “true boundary” (Definition
2). In order to optimize these algorithms, we also propose
a techn ique, that we call relaxation, which can simplify the
decomposition process and reduce the number of redundant
compon ents in the output. Moreover, we present procedures
for basic set-theoretical operations on semi-algebraic sets
represented by triangular d ecomposition. Experimentation
confirms the effectiveness of our techniques.
1. INTRODUCTION
Triangular decompositions of semi-algebraic systems were
introduced in [3]. The key notions and notations of this
paper are reviewed in the next section.
That paper presents also an algorithm for generating those
decompositions. This algorithm can either be eager, com-
puting the entire decomposition, or lazy, only computing
the decomposition corresponding to the highest ( complex )
dimensional components, and deferring lower-dimensional
compon ents. While a complete decomposition is known to
have a worst-case complexity which is doubly-exponential in
the number of variables [8], under plausible assumptions the
lazy variant has a singly-exponential complexity. Neverthe-
less, it is still desirable to improve the practical efficiency of
both types of decomposition.
The notion of a border polynomial [15] is at the core of our
Rong Xiao
University of Western Ontario
rong@csd.uwo.ca
work. A strongly related n otion, discriminant variety, was
introduced in [9] and the link between them was investi-
gated in [14]. Other similar but more restrictive notions
like “generalised discriminant” and “generalised resultant”
were introduced in [10]. For a squarefree regular chain T ,
regarded as a real parametric system in its free variables u,
the border polynomial BP (T ) encodes th e locus of the u-
values at which T has lower rank or at which T is no longer
a squarefree regular chain. (See §2 for the notions related to
triangular decomposition and regular chains.) Consequently,
for each connected component C of the complement of t he
real hypersurface defined by BP (T ) the number of real so-
lutions of the regular chain T is constant at any point of C.
However, BP (T ) is not an invariant of the variety
W (T ),
which is a bottleneck in designing better algorithms based
on the notion of a border polynomial. We overcome this
difficulty in two ways.
Firstly, in §3, we prove that among all regular chains T
satisfying sat(T
) = sat(T ) there is one and only one (char-
acterized in Theorem 1) for which BP (T ) is minimal w.r.t.
inclusion. Secondly, in §4, we introduce the concept of an
effective boundary of a squarefree semi-algebraic system, see
Definition 2. This allows us to identify a subset of BP (T )
which is an invariant of
W (T ), that is, unchanged when re-
placing T by T as long as
W (T ) = W (T
) holds. In many
ways, our notion of effective boundary is similar to the “bet-
ter projection” ideas in the classical [7, and many others]
approach to cylindrical algebraic decomposition.
In §5, we introduce the technique of relaxation which we
shall motivate by an example. Consider the semi-algebraic
system sys = [f = 0, x − b > 0], where f = ax
3
+ bx − a for
the variable ordering a < b < x. The LazyRealTriangularize
algorithm of [3] will compute the border polynomial set B =
{a, b
1
, b
2
} and the fingerprint polynomial set (FPS) F =
{a, b
1
, b
2
, b, p
1
, p
2
, p
3
}. where b
1
= ab
3
+ b
2
− a, b
2
= 27a
3
+
4b
3
, p
1
= 2b
3
+ 1, p
2
= b
3
− 4 and p
3
= b − 1. Thus
the LazyRealTriangularize(sys) will produce 1 regular semi-
algebraic system S
1
= [Q
1
, {f = 0, x − b > 0}], and 7 un-
evaluated recursive calls, where
�
�
�
6 6 =
�
�
�
Q
1
= (b < 0 ∧ p
1
= 6 0 ∧ a 6 0 ∧ b
2
=6 0 ∧ b
1
= = 6 0)
(p
1
> 0 ∧ b
1
> 0 ∧ a < 0 ∧ p
3
> 0 ∧ p
2
6= 0 ∧ b
2
6 0) =
(b > 0 ∧ p
1
> 0 ∧ b
1
6 0 ∧ a < 0 ∧ p
3
< 0 ∧ p
2
< 0 ∧ b
2
6 0) = =
(b > 0 ∧ p
1
> 0 ∧ b
1
< 0 ∧ a > 0 ∧ p
3
< 0 ∧ p
2
< 0 ∧ b
2
> 0)
and the 7 calls are made for each p ∈ F with the form
LazyRealTriangularize([p = 0, f = 0, x − b > 0]). The key
observation is that some of these recursive calls can simply
be avoided if some of the strict ineq ualities in Q
1
can be
relaxed, that is, replaced by non-strict inequalities. The
results of §5, and in particular Theorem 5 provide criteria
for this p urpose. Returning to our example, when relaxation
techniques are used LazyRealTriangularize(sys) will produce
1 regular semi-algebraic system S
2
= [Q
2
, {f = 0, x− b > 0}],
and 3 un-evaluated recursive calls, where
Q
2
= (b ≤ 0 ∧ b
1
= 0 ∧ a = 0 ∧ b
2
6 0)
(p
1
≥ 0 ∧ b
1
> 0 ∧ a < 0 ∧ p
3
≥ 0 ∧ b
2
6= 0)
(b ≥ 0 ∧ p
1
≥ 0 ∧ b
1
6 0 ∧ 6 0) = a < 0 ∧ p
3
≤ 0 ∧ p
2
≤ 0 ∧ b
2
=
(b ≥ 0 ∧ p
1
≥ 0 ∧ b
1
< 0 ∧ a > 0 ∧ p
3
≤ 0 ∧ p
2
≤ 0 ∧ b
2
> 0)
Moreover, it turns that the the 3 un-evaluated recursive calls
are of the form LazyRealTriangularize([p = 0, f = 0, x − b >
0]), for p ∈ B. Continuing with that example, one can check
that the full triangular decomposition of sys produces 16
and 9 regular semi-algebraic systems, without and with re-
laxation techniques, respectively. Therefore, relaxation t ech-
niques can help simplify the output of our algorithms.
Nevertheless, even with relaxation techniques, our algorithms
can produce redundant components, that is, a regular semi-
algebraic system S for which there exists another regular
semi-algebraic system S
in the same decomposition and
such that Z
R
(S) ⊆ Z
R
(S
) holds. This is actually the case
for our example where 1 out of the 9 regular semi-algebraic
systems is redundant.
To perform inclusion test on the zero sets of regular semi-
algebraic systems, we have developed algorithms for set-
theoretical operations on semi-algebraic sets represented by
triangular decomposition, see §7. Those algorithms rely on
a new algorithm, presented in §6, for computing triangular
decomposition of semi-algebraic systems in an incremental
manner, which is a natural adaption of the idea presented
in [11] for computing triangular decomposition of algebraic
systems incrementally.
The experimentation illustrates the effectiveness of the dif-
ferent techniques presented in this p aper. In particular, we
observe that with relaxation, the decomposition algorithm
will produce output with less redundancy without paying a
lot, and accelerate on some hard systems; the incremental
algorithm for computin g triangular decomposition of semi-
algebraic systems often ou tperforms the one in [3]. More-
over, we observe that our techniques for removing redundant
compon ents can usually process in a “reasonable” amount
time the outpu t of the systems that RealTriangularize can
decompose.
2. TRIANGULAR DECOMPOSITION
We summarize below the notions and notations of [3], includ-
ing triangular decompositions of semi-algebraic systems.
Zero sets and topology. In this paper, we use “Z” to denote
the zero set of a polynomial system, involving equations and
inequations, in C
n
and “Z
R
” to denote the zero set of a semi-
algebraic system in R
n
. If a semi-algebraic set S is finite, we
denote by #(S) the number of distinct points in it. In R
n
,
we use the Euclidean topology; in C
n
, we use the Zariski
topology. Given a semi-algebraic set S, we denote by ∂ S
the boundary of S, by
S the closure of S.
Notations on polynomials. Throughout t his paper, all poly-
nomials are in Q[x], with ordered variables x = x
1
< · · · <
x
n
. We order monomials of Q[x] by the lexicographical or-
dering induced by x
1
< · · · < x
n
. Then, we require that th e
leading coefficient of every polynomial in a regular chain or
in a border polynomial set (defined hereafter) is equal to 1.
Let F ⊂ Q[x]. We denote by V (F ) the set of common zeros
of F in C
n
. Let p be a polynomial in Q[x] \ Q. Then denote
by mvar(p), init(p), and mdeg(p) respectively t he greatest
variable appearing in p (called the main variable of p), the
leading coefficient of p w.r.t. mvar(p) (called th e initial of p) ,
and the degree of p w.r.t. mvar(p) (called the main degree
of p). Let v ∈ x. Denote by lc(p, v), deg(p, v), der(p, v),
discrim(p, v) respectively the leading coefficient, the degree,
the derivative and the discriminant of p w.r.t. v.
Triangular set. Let T ⊂ Q[x] be a triangular set, that is, a
set of non-constant polynomials with pairwise distinct main
variables. Denote by mvar(T ) the set of main variables of
the polynomials in T . A variable v in x is called algebraic
w.r.t. T if v ∈ mvar(T ), otherwise it is said free w.r.t. T .
If no confusion is possible, we shall always denote by u =
u
1
, . . . , u
d
and y = y
1
, . . . , y
m
(m + d = n) respectively the
free and the main variables of T . When T is regarded as a
parametric system, the free variables in T are its parameters.
Let h
T
be the product of the initials of the polynomials
in T . We denote by sat(T ) the saturated ideal of T : if T
is the empty triangular set, then sat(T ) is defined as the
trivial ideal 0, otherwise it is the colon ideal T : h
∞
.
T
The quasi-component W (T ) of T is defined as V (T ) \ V (h
T
).
Denote by
W (T ) the Zariski closure of W (T ), which is equal
to V (sat(T )). Den ote by W
R
(T ) the set Z
R
(T ) \ Z
R
(h
T
).
Iterated resultant. Let p, q ∈ Q[x] \ Q. Let v = mvar(q).
Denote by res(p, q, v) the resultant of p, q w.r.t. v. Let
T ⊂ Q[x] be a triangular set. We define res(p, T ) ind uc-
tively: if T is empty, then res(p, T ) = p; otherwise let
v be the largest variable occurring in T , then res(p, T ) =
res(res(p, T
v
, v), T
<v
), where T
v
and T
<v
denote respectively
the polynomials of T with main variables equal to and less
than v.
Regular chain. A triangular set T ⊂ Q[x] is called a regular
chain if: either T is empty; or (letting t be the polynomial
in T with maximum main variable), T \ {t} is a regular
chain, and the initial of t is regular w.r.t. sat(T \ {t}). Let
H ⊂ Q[x]. The pair [T, H] is a regular system if each poly-
nomial in H is regular modulo sat(T ). A regular chain T or
a regular system [T, H], is squarefree if for all t ∈ T , der(t) is
regular w.r.t. sat(T ). Given u ∈ R
d
, we say that a squarefree
regular system [T, H] specializes well at u if h
T
(u) 0 an d=
[T (u), H(u)] is a squarefree regular system. A regular chain
is called d-dimensional if it has d free variables.
Semi-algebraic system. Consider four finite polynomial sets
F = {f
1
, . . . , f
s
}, N = {n
1
, . . . , n
k
}, P = {p
1
, . . . , p
e
}, and
H = {h
1
, . . . , h
ℓ
} of Q[x]. Let N
≥
denote the set of non-
negative inequalities {n
1
≥ 0, . . . , n
k
≥ 0}. Let P
>
denote
the set of positive inequalities {p
1
> 0, . . . , p
e
> 0}. Let
H
denote the set of inequations {h
1
= 0}.
=
0, . . . , h
ℓ
= We
denote by S = [F, N
≥
, P
>
, H
] the semi-algebraic system
=
(SAS) defined as the conjunction of the constraints f
1
=
· · · f
s
= 0, N
≥
, P
>
, H
=
. When N
≥
, H
are empty, S is
=
called a basic semi -algebraic system and denoted by [F, P
>
].
Regular semi-algebraic system. We call a basic SAS [T, P
>
]
in Q[u, y] a squarefree semi-algebraic system, SFSAS for
short, if [T, P ] forms a squarefree regular system. Let [T, P
>
]
be an SFSAS. Let Q be a quantifier-free formula of Q[u]. We
say that R := [Q, T, P
>
] is a regular semi-algebraic system if
(i) Q defines a non-empty op en semi-algebraic set S in R
d
;
(ii) [T, P ] specializes well at every point of S,
(iii) at each u ∈ S, the specialized system [T (u), P (u)
>
]
has at least one real zero.
Border polynomial [15, 16, 3]. We review briefly the notion
of border polynomial of a regular chain, a regular system, or
an SFSAS . Let R be either a squarefree regular chain T , or
a squarefree regular system [T, P ], or an SFSAS [T, P
>
] in
Q[x]. We denote by B
sep
(T ), B
ini
(T ), B
ineqs
([T, P ]) the set
of irreducible factors of:
t∈T
res(discrim(t, mvar(t)), T ),
t∈T
and
f ∈P
res(f, T ), respectively. Denote by BP(R) the set
B
sep
(T ) ∪ B
ini
(T ) ∪ B
ineqs
([T, P ]). Then BP(R) (resp. the
polynomial
f ∈BP(R)
f) is called the border polynomial set
(resp. border polynomial) of R.
Lemma 1 (Lemma 2 in [3]). Let R = [T, P
>
] be an SF-
SAS of Q[x]. Let u
1
, u
2
be two parameter values in a same
connected component of Z
R
(
f ∈BP(R)
f = 0) i n R
d
. Then
#Z
R
(R(u
1
)) = #Z
R
(R(u
2
)).
Fingerprint polynomial set. R = [B
, T, P
>
] is called a
=
pre-regular semi-algebraic system, if for each p ∈ BP([T, P
>
]),
p is a factor of some polynomial in B. Suppose R is a pre-
regular semi-algebraic system. A polynomial set D ⊂ Q[u]
is called a fingerprint polynomial set (FPS) of R if:
(i) Z
R
(D
=
) ⊆ Z
R
(B
=
) holds,
(ii) for all α, β ∈
) with α β, if the signs of p(α)Z
R
(D
=
=
and p(β) are the same for all p ∈ D, then R(α) has
real solutions if and only if R(β) does.
Open CAD operator [12, 2, 3]. Let u = u
1
< · · · < u
d
be ordered variables. For a polynomial p ∈ Q[u], denote by
factor(p) the set of the non-constant irreducible factors of
p; for A ⊂ Q[u], define factor(A) = ∪
p∈A
factor(p). For a
squarefree polynomial p, the open projection operator (oproj)
w.r.t. a variable v ∈ u is defined as below:
oproj(p, v) := factor(discrim(p, v) lc(p, v)).
∗
If p is not squarefree, then we define oproj(p, v) := oproj(p , v),
∗
where p is the squarefree part of p; then for a polyn omial
set A, we define oproj(A, v) := oproj(Π
f ∈A
f, v).
Given A ⊂ Q[u] and x ∈ {u
1
, . . . , u
d
}, denote by der(A, x)
the derivative cl osure of A w.r.t. x. The open augmented
projected factors of A, denoted by oaf(A), is defined as
follows. Let k be the smallest positive integer such that
A ⊂ Q[u
1
, . . . , u
k
] holds. Let C = factor(der(A, u
k
)); we
have:
1. if k = 1, then oaf(A) := C;
2. if k > 1, then oaf(A) := C ∪ oaf(oproj(C, u
k
)).
3. BORDER POLYNOMIAL
The relation “having the same saturated ideal” is an eq uiva-
lence relation among regular chains of Q[x]. We show in this
section that, for each equivalence class, there exists a unique
representative whose border polynomial set is contained in
the border polynomial set of any oth er representative.
To this end, we rely on the concept of canonical regular
chain. In the field of triangular decompositions, several au-
thors have used this term to refer to different notions. To
be precise, we make use of the one defined in [13].
Definition 1 (canonical regular chain). Let T be
a regular chain of Q[x]. If each polynomial t of T satisfies:
1. the initial of t involves only the free variables of T ,
2. for any polynomial f ∈ T with mvar(f) < mvar(t), we
have deg(t, mvar(f )) < mdeg(f),
res(init(t
t
), T ),
3. is primitive over Q, w.r.t. its main variable,
then we say that T is canonical.
Remark 1. Let T = {t
1
, · · · , t
m
} be a regular chain; let
d
k
= mdeg(t
k
), for k = 1 · · · m . One constructs a canon-
∗ ∗ ∗ ∗
ical regular chain T = {t
1
, t
2
, . . . , t
m
} such that sat(T ) =
∗
sat(T ) in the following way:
∗
1. set t
1
to be the primitive part of t
1
w.r.t. y
1
;
2. for k = 2, . . . , m, let r
k
be the iterated resultant
res(init(t
k
), {t
1
, . . . , t
k−1
}). Suppose r
k
= a
k
init(t
k
) +
k−1
c
i
t
i
. Compute t as the pseudo-reminder of a
k
t
k
+
i=1
(
k−1
c
i
t
i
)y
d
k
by {t
∗
1
, . . . , t
∗
k−1
}. Set t
∗
k
to be the prim-
i=1
k
itive part of t w.r.t. y
k
.
A canon ical regular chain has the minimal border polyno-
mial set among the family of regular chains having the same
saturated ideal, which is stated in the following theorem.
Theorem 1. Given a squarefree regular chains T of Q[x],
∗
there exists a unique canonical regular chain T such that
sat(T ) = sat(T
∗
). Moreover, we have BP(T
∗
) ⊆ BP(T ).
The proof of the above theorem relies on some basic prop-
erties of border polynomial set recalled below.
Given a constructible set C defined by a parametric poly-
nomial system, the minimal discriminant variety (MDV) [9]
of C, denoted by mdv(C), is an intrinsic geometric object
attached to C and the parameters. The following results re-
late the border polynomial of a regular chains T and the
discriminant variety of the algebraic variety V (T ).
Lemma 2 ([14]). Let T be a squarefree regular chain of
Q[u, y]. Then we have mdv(V (T )) = V (
f ∈BP(T )
f).
Lemma 3 ([14, Lemma 17]). Let T be a squarefree reg-
ular chain of Q[u, y]. T hen we have mdv(
W (T )) ⊆ mdv(V (T ))
and mdv(V (T )) \ mdv(
W (T )) ⊆ V (
f).
f ∈B
ini
(T )
Lemma 4. Let T
1
and T
2
be squarefree regular chains of
Q[x] such that sat(T
1
) = sat(T
2
). If B
ini
(T
1
) ⊆ B
ini
(T
2
),
then we have BP(T
1
) ⊆ BP(T
2
).
Proof. Firstly, we have V (
f) ⊆ mdv(V (T
i
))
f ∈B
ini
(T
i
)
by Lemma 2. Then with Lemma 3, we have mdv(V (T
i
)) =
V (
f) ∪ mdv(
W (T
i
)). Since sat(T
1
) = sat(T
2
),
f ∈B
ini
(T
i
)
we have W (T
1
) = W (T
2
). Therefore we have mdv(V (T
2
)) ⊆
mdv(V (T
1
)) by the assumpt ion B
ini
(T
1
) ⊆ B
ini
(T
2
), which
implies the lemma.
Next we prove Theorem 1.
Proof. By Remark 1, we can always construct a canoni-
∗ ∗
cal regular chain T such that sat(T ) = sat(T ). Moreover,
∗
for each t ∈ T , we have init(t ) divides res(init(t), T ). There-
∗ ∗
fore, B
ini
(T ) ⊆ B
ini
(T ) holds, which implies BP(T ) ⊆
BP(T ) by Lemma 4.
Suppose T
is any given canonical regular chain such that
sat(T
) = sat(T ) holds. It is sufficient to show that T
∗
= T
holds to complete the proof.
Note that T
, T
∗
and T have the same set of free and al-
gebraic variables, denoted respectively by u and y. Given
I an ideal in Q[u, y], denote by I
ext
the extension of I in
Q(u)[y]. Since p
ext
= 1 holds for any prime ideal p in
Q[u, y] with u algebraically dependent, we have T
∗
ext
=
T
ext
= sat(T )
ext
holds. Therefore, th e polynomials in T
∗
(or T
) form a Gr¨obner basis of sat(T )
ext
(w.r.t. the lexico-
graphical ordering on y) since their leading power products
∗
are pairwise coprime. Dividing each polynomial in T (or
T
) by its initial, we obtain the unique reduced Gr¨obner
basis of sat(T )
ext
. This implies T
∗
= T
.
4. EFFECTIVE BOUNDARY AND FPS
In this subsection, we will focus on an SFSAS S = [T, P
>
]
in Q[u, y] where u = u
1
, . . . , u
d
are the free variables of T .
Definition 2 (Effective boundary). Let h be a (d−
1)-dimensional hypersurface in the parameter space R
d
of S.
We call h an effective boundary of S if for every hypersur-
face H ⊇ h in R
d
, there exists a point u
∗
in h \ H satisfying:
∗ ∗
for any open ball O(u ) of u , there exist two points α
1
,
∗
α
2
∈ O(u ) \ h, s.t. #Z
R
(S(α
1
)) = #Z
R
(S(α
2
)). Denote
by E(S) the union of all effective boundaries of S.
Recall that the hypersurface defined by the border polyno-
mial of an SFSAS partitions the parametric space into re-
gions, where the number of real solutions is locally invariant.
One might imagine that the effective boundaries are strongly
related to the border polynomial set. Indeed, we have the
following Lemma stating the relation.
Lemma 5. We have E(S) ⊆ Z
R
(
f ∈BP(S)
f = 0).
Proof. Let h ∈ E(S) such that h ⊆ Z
R
(
f ∈BP(S)
f =
0) holds. Then for each u ∈ h \ Z
R
(
f ∈BP(S)
f = 0), we
can choose an open ball O(u) of u contained in a connected
compon ent of the set Z
R
(
f ∈B
f 0). By Lemma 1, for =
any two points α
1
, α
2
∈ O(u), #Z
R
(S(α
1
)) = #Z
R
(S(α
2
))
holds. That is a contradiction to the assumption of h being
an effective boundary.
Lemma 5 implies that the set of effective boun daries repre-
sented by irreducible polynomials of Q[u] is finite and can
be given by polynomials from the border polynomial set.
Definition 3. A polynomial p in BP(S) is called an ef-
fective border polynomial factor if Z
R
(p = 0) is an effective
boundary of S. We denote by ebf(S) the set of effective
border polynomial factors.
The example below shows that some of the polynomials in a
border polynomial may not be effective. Roughly speaking,
the factors in B
ini
are not effective. This property is formally
stated in a soon coming extended version of this article.
Example 1. Consider an SFSAS R = [{ax
2
+bx+1}, { }].
Its border polynomial set is {a, b
2
−4a}. One can verify from
Figure 1 that Z
R
(b
2
− 4a = 0) is an effective boundary of R,
while Z
R
(a = 0) is not. Indeed, all a, b-values in the blank
(resp, filled) area specialize R to have 2 (resp. 0) real solu-
tions.
Figure 1: Effective and non-effective boundary
Since E(S) can be described by border polynomial factors,
we derive the following theorem, which can be viewed as a
“computable-version” of Defin ition 2.
