<s>
In	O
the	O
important	O
case	O
of	O
univariate	B-General_Concept
polynomials	O
over	O
a	O
field	O
the	O
polynomial	O
GCD	O
may	O
be	O
computed	O
,	O
like	O
for	O
the	O
integer	O
GCD	O
,	O
by	O
the	O
Euclidean	O
algorithm	O
using	O
long	O
division	O
.	O
</s>
<s>
The	O
similarity	O
between	O
the	O
integer	O
GCD	O
and	O
the	O
polynomial	O
GCD	O
allows	O
extending	O
to	O
univariate	B-General_Concept
polynomials	O
all	O
the	O
properties	O
that	O
may	O
be	O
deduced	O
from	O
the	O
Euclidean	O
algorithm	O
and	O
Euclidean	O
division	O
.	O
</s>
<s>
They	O
are	O
a	O
fundamental	O
tool	O
in	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
because	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
use	O
them	O
systematically	O
to	O
simplify	O
fractions	O
.	O
</s>
<s>
Conversely	O
,	O
most	O
of	O
the	O
modern	O
theory	O
of	O
polynomial	O
GCD	O
has	O
been	O
developed	O
to	O
satisfy	O
the	O
need	O
for	O
efficiency	O
of	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
.	O
</s>
<s>
For	O
univariate	B-General_Concept
polynomials	O
over	O
a	O
field	O
,	O
one	O
can	O
additionally	O
require	O
the	O
GCD	O
to	O
be	O
monic	O
(	O
that	O
is	O
to	O
have	O
1	O
as	O
its	O
coefficient	O
of	O
the	O
highest	O
degree	O
)	O
,	O
but	O
in	O
more	O
general	O
cases	O
there	O
is	O
no	O
general	O
convention	O
.	O
</s>
<s>
For	O
two	O
univariate	B-General_Concept
polynomials	O
and	O
over	O
a	O
field	O
,	O
there	O
exist	O
polynomials	O
and	O
,	O
such	O
that	O
and	O
divides	O
every	O
such	O
linear	O
combination	O
of	O
and	O
(	O
Bézout	O
's	O
identity	O
)	O
.	O
</s>
<s>
There	O
are	O
several	O
ways	O
to	O
find	O
the	O
greatest	B-Algorithm
common	I-Algorithm
divisor	I-Algorithm
of	I-Algorithm
two	I-Algorithm
polynomials	I-Algorithm
.	O
</s>
<s>
If	O
the	O
coefficients	O
are	O
floating-point	B-Algorithm
numbers	I-Algorithm
that	O
represent	O
real	O
numbers	O
that	O
are	O
known	O
only	O
approximately	O
,	O
then	O
one	O
must	O
know	O
the	O
degree	O
of	O
the	O
GCD	O
for	O
having	O
a	O
well	O
defined	O
computation	O
result	O
(	O
that	O
is	O
a	O
numerically	B-Algorithm
stable	I-Algorithm
result	O
;	O
in	O
this	O
cases	O
other	O
techniques	O
may	O
be	O
used	O
,	O
usually	O
based	O
on	O
singular	O
value	O
decomposition	O
.	O
</s>
<s>
The	O
case	O
of	O
univariate	B-General_Concept
polynomials	O
over	O
a	O
field	O
is	O
especially	O
important	O
for	O
several	O
reasons	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
univariate	B-General_Concept
polynomials	O
over	O
a	O
field	O
,	O
it	O
may	O
be	O
stated	O
as	O
follows	O
.	O
</s>
<s>
The	O
elements	O
of	O
are	O
usually	O
represented	O
by	O
univariate	B-General_Concept
polynomials	O
over	O
of	O
degree	O
less	O
than	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
univariate	B-General_Concept
polynomials	O
,	O
there	O
is	O
a	O
strong	O
relationship	O
between	O
the	O
greatest	O
common	O
divisors	O
and	O
resultants	O
.	O
</s>
<s>
Every	O
coefficient	O
of	O
the	O
subresultant	O
polynomials	O
is	O
defined	O
as	O
the	O
determinant	O
of	O
a	O
submatrix	O
of	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
of	O
P	O
and	O
Q	O
.	O
</s>
<s>
be	O
two	O
univariate	B-General_Concept
polynomials	O
with	O
coefficients	O
in	O
a	O
field	O
K	O
.	O
Let	O
us	O
denote	O
by	O
the	O
K	O
vector	O
space	O
of	O
dimension	O
i	O
of	O
polynomials	O
of	O
degree	O
less	O
than	O
i	O
.	O
</s>
<s>
The	O
resultant	O
of	O
P	O
and	O
Q	O
is	O
the	O
determinant	O
of	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
,	O
which	O
is	O
the	O
(	O
square	O
)	O
matrix	O
of	O
on	O
the	O
bases	O
of	O
the	O
powers	O
of	O
X	O
.	O
</s>
<s>
First	O
we	O
add	O
(	O
i	O
+	O
1	O
)	O
columns	O
of	O
zeros	O
to	O
the	O
right	O
of	O
the	O
(	O
m	O
+	O
n	O
−	O
2i	O
−	O
1	O
)	O
×	O
(	O
m	O
+	O
n	O
−	O
2i	O
−	O
1	O
)	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
implies	O
that	O
the	O
submatrix	O
of	O
the	O
m	O
+	O
n	O
−	O
2i	O
first	O
rows	O
of	O
the	O
column	O
echelon	O
form	O
of	O
Ti	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
and	O
thus	O
that	O
si	O
is	O
not	O
0	O
.	O
</s>
<s>
For	O
univariate	B-General_Concept
polynomials	O
over	O
the	O
rational	O
numbers	O
,	O
one	O
may	O
think	O
that	O
Euclid	O
's	O
algorithm	O
is	O
a	O
convenient	O
method	O
for	O
computing	O
the	O
GCD	O
.	O
</s>
<s>
A	O
polynomial	O
in	O
n	O
variables	O
may	O
be	O
considered	O
as	O
a	O
univariate	B-General_Concept
polynomial	O
over	O
the	O
ring	O
of	O
polynomials	O
in	O
(	O
n	O
−	O
1	O
)	O
variables	O
.	O
</s>
<s>
The	O
proof	O
that	O
a	O
polynomial	O
ring	O
over	O
a	O
unique	O
factorization	O
domain	O
is	O
also	O
a	O
unique	O
factorization	O
domain	O
is	O
similar	O
,	O
but	O
it	O
does	O
not	O
provide	O
an	O
algorithm	O
,	O
because	O
there	O
is	O
no	O
general	O
algorithm	O
to	O
factor	O
univariate	B-General_Concept
polynomials	O
over	O
a	O
field	O
(	O
there	O
are	O
examples	O
of	O
fields	O
for	O
which	O
there	O
does	O
not	O
exist	O
any	O
factorization	O
algorithm	O
for	O
the	O
univariate	B-General_Concept
polynomials	O
)	O
.	O
</s>
<s>
Given	O
two	O
polynomials	O
A	O
and	O
B	O
in	O
the	O
univariate	B-General_Concept
polynomial	O
ring	O
Z[X],	O
the	O
Euclidean	O
division	O
(	O
over	O
Q	O
)	O
of	O
A	O
by	O
B	O
provides	O
a	O
quotient	O
and	O
a	O
remainder	O
which	O
may	O
not	O
belong	O
to	O
Z[X]	O
.	O
</s>
<s>
However	O
,	O
modern	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
only	O
use	O
it	O
if	O
F	O
is	O
finite	O
because	O
of	O
a	O
phenomenon	O
called	O
intermediate	B-Algorithm
expression	I-Algorithm
swell	I-Algorithm
.	O
</s>
