<s>
In	O
mathematics	O
,	O
a	O
Sylvester	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
associated	O
to	O
two	O
univariate	O
polynomials	O
with	O
coefficients	O
in	O
a	O
field	O
or	O
a	O
commutative	O
ring	O
.	O
</s>
<s>
The	O
entries	O
of	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
of	O
two	O
polynomials	O
are	O
coefficients	O
of	O
the	O
polynomials	O
.	O
</s>
<s>
The	O
determinant	O
of	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
of	O
two	O
polynomials	O
is	O
their	O
resultant	O
,	O
which	O
is	O
zero	O
when	O
the	O
two	O
polynomials	O
have	O
a	O
common	O
root	O
(	O
in	O
case	O
of	O
coefficients	O
in	O
a	O
field	O
)	O
or	O
a	O
non-constant	O
common	O
divisor	O
(	O
in	O
case	O
of	O
coefficients	O
in	O
an	O
integral	O
domain	O
)	O
.	O
</s>
<s>
The	O
Sylvester	B-Algorithm
matrix	I-Algorithm
associated	O
to	O
p	O
and	O
q	O
is	O
then	O
the	O
matrix	B-Architecture
constructed	O
as	O
follows	O
:	O
</s>
<s>
Thus	O
,	O
if	O
m	O
=	O
4	O
and	O
n	O
=3	O
,	O
the	O
matrix	B-Architecture
is	O
:	O
</s>
<s>
If	O
one	O
of	O
the	O
degrees	O
is	O
zero	O
(	O
that	O
is	O
,	O
the	O
corresponding	O
polynomial	O
is	O
a	O
nonzero	O
constant	O
polynomial	O
)	O
,	O
then	O
there	O
are	O
zero	O
rows	O
consisting	O
of	O
coefficients	O
of	O
the	O
other	O
polynomial	O
,	O
and	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
dimension	O
the	O
degree	O
of	O
the	O
non-constant	O
polynomial	O
,	O
with	O
the	O
all	O
diagonal	O
coefficients	O
equal	O
to	O
the	O
constant	O
polynomial	O
.	O
</s>
<s>
If	O
m	O
=	O
n	O
=	O
0	O
,	O
then	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
is	O
the	O
empty	O
matrix	B-Architecture
with	O
zero	O
rows	O
and	O
zero	O
columns	O
.	O
</s>
<s>
The	O
above	O
defined	O
Sylvester	B-Algorithm
matrix	I-Algorithm
appears	O
in	O
a	O
Sylvester	O
paper	O
of	O
1840	O
.	O
</s>
<s>
In	O
a	O
paper	O
of	O
1853	O
,	O
Sylvester	O
introduced	O
the	O
following	O
matrix	B-Architecture
,	O
which	O
is	O
,	O
up	O
to	O
a	O
permutation	O
of	O
the	O
rows	O
,	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
of	O
p	O
and	O
q	O
,	O
which	O
are	O
both	O
considered	O
as	O
having	O
degree	O
max(m, n )	O
.	O
</s>
<s>
This	O
is	O
thus	O
a	O
-matrix	O
containing	O
pairs	O
of	O
rows	O
.	O
</s>
<s>
Thus	O
,	O
if	O
m	O
=	O
4	O
and	O
n	O
=3	O
,	O
the	O
matrix	B-Architecture
is	O
:	O
</s>
<s>
The	O
determinant	O
of	O
the	O
1853	O
matrix	B-Architecture
is	O
,	O
up	O
to	O
sign	O
,	O
the	O
product	O
of	O
the	O
determinant	O
of	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
(	O
which	O
is	O
called	O
the	O
resultant	O
of	O
p	O
and	O
q	O
)	O
by	O
(	O
still	O
supposing	O
)	O
.	O
</s>
<s>
In	O
such	O
a	O
case	O
,	O
the	O
determinant	O
of	O
the	O
associated	O
Sylvester	B-Algorithm
matrix	I-Algorithm
(	O
which	O
is	O
called	O
the	O
resultant	O
of	O
the	O
two	O
polynomials	O
)	O
equals	O
zero	O
.	O
</s>
<s>
This	O
means	O
the	O
kernel	B-Algorithm
of	O
the	O
transposed	O
Sylvester	B-Algorithm
matrix	I-Algorithm
gives	O
all	O
solutions	O
of	O
the	O
Bézout	O
equation	O
where	O
and	O
.	O
</s>
<s>
Consequently	O
the	O
rank	O
of	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
determines	O
the	O
degree	O
of	O
the	O
greatest	B-Algorithm
common	I-Algorithm
divisor	I-Algorithm
of	O
p	O
and	O
q	O
:	O
</s>
<s>
Moreover	O
,	O
the	O
coefficients	O
of	O
this	O
greatest	B-Algorithm
common	I-Algorithm
divisor	I-Algorithm
may	O
be	O
expressed	O
as	O
determinants	O
of	O
submatrices	O
of	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
(	O
see	O
Subresultant	O
)	O
.	O
</s>
