<s>
In	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
a	O
triangular	B-Algorithm
decomposition	I-Algorithm
of	O
a	O
polynomial	O
system	O
is	O
a	O
set	O
of	O
simpler	O
polynomial	O
systems	O
such	O
that	O
a	O
point	O
is	O
a	O
solution	O
of	O
if	O
and	O
only	O
if	O
it	O
is	O
a	O
solution	O
of	O
one	O
of	O
the	O
systems	O
.	O
</s>
<s>
If	O
the	O
coefficients	O
of	O
the	O
polynomial	O
systems	O
are	O
real	O
numbers	O
,	O
then	O
the	O
real	O
solutions	O
of	O
can	O
be	O
obtained	O
by	O
a	O
triangular	B-Algorithm
decomposition	I-Algorithm
into	O
regular	O
semi-algebraic	O
systems	O
.	O
</s>
<s>
There	O
are	O
various	O
algorithms	O
available	O
for	O
obtaining	O
triangular	B-Algorithm
decomposition	I-Algorithm
of	O
both	O
in	O
the	O
sense	O
of	O
Kalkbrener	O
and	O
in	O
the	O
sense	O
of	O
Lazard	O
and	O
Wen-Tsun	O
Wu	O
.	O
</s>
<s>
The	O
Lextriangular	O
Algorithm	O
by	O
Daniel	O
Lazard	O
and	O
the	O
Triade	O
Algorithm	O
by	O
together	O
with	O
the	O
Characteristic	O
Set	O
Method	O
are	O
available	O
in	O
various	O
computer	B-Algorithm
algebra	I-Algorithm
systems	O
,	O
including	O
Axiom	B-Language
and	O
Maple	B-Language
.	O
</s>
<s>
For	O
,	O
regarded	O
as	O
a	O
system	O
of	O
polynomial	O
equations	O
,	O
there	O
are	O
two	O
notions	O
of	O
a	O
triangular	B-Algorithm
decomposition	I-Algorithm
over	O
the	O
algebraic	O
closure	O
of	O
.	O
</s>
<s>
The	O
regular	O
semi-algebraic	O
systems	O
form	O
a	O
triangular	B-Algorithm
decomposition	I-Algorithm
of	O
the	O
semi-algebraic	O
system	O
.	O
</s>
<s>
According	O
to	O
the	O
Maple	B-Language
code	O
:	O
One	O
possible	O
triangular	B-Algorithm
decompositions	I-Algorithm
of	O
the	O
solution	O
set	O
of	O
with	O
using	O
library	O
is	O
:	O
</s>
