In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. The notion of a regular semi-algebraic system is an adaptation of the concept of a regular chain focusing on solutions of the real analogue: semi-algebraic systems.
Any semi-algebraic system
S
can be decomposed into finitely many regular semi-algebraic systems
S
1
,
…
,
S
e
such that a point (with real coordinates) is a solution of
S
if and only if it is a solution of one of the systems
S
1
,
…
,
S
e
.
Let
T
be a regular chain of
k
[
x
1
,
…
,
x
n
]
for some ordering of the variables
x
=
x
1
,
…
,
x
n
and a real closed field
k
. Let
u
=
u
1
,
…
,
u
d
and
y
=
y
1
,
…
,
y
n
−
d
designate respectively the variables of
x
that are free and algebraic with respect to
T
. Let
P
⊂
k
[
x
]
be finite such that each polynomial in
P
is regular w.r.t.\ the saturated ideal of
T
. Define
P
>
:=
{
p
>
0
∣
p
∈
P
}
. Let
Q
be a quantifier-free formula of
k
[
x
]
involving only the variables of
u
. We say that
R
:=
[
Q
,
T
,
P
>
]
is a regular semi-algebraic system if the following three conditions hold.
Q
defines a non-empty open semi-algebraic set
S
of
k
d
,
the regular system
[
T
,
P
]
specializes well at every point
u
of
S
,
at each point
u
of
S
, the specialized system
[
T
(
u
)
,
P
(
u
)
>
]
has at least one real zero.
The zero set of
R
, denoted by
Z
k
(
R
)
, is defined as the set of points
(
u
,
y
)
∈
k
d
×
k
n
−
d
such that
Q
(
u
)
is true and
t
(
u
,
y
)
=
0
,
p
(
u
,
y
)
>
0
, for all
t
∈
T
and all
p
∈
P
. Observe that
Z
k
(
R
)
has dimension
d
in the affine space
k
n
.
