<s>
In	O
commutative	O
algebra	O
and	O
algebraic	O
geometry	O
,	O
elimination	B-Algorithm
theory	I-Algorithm
is	O
the	O
classical	O
name	O
for	O
algorithmic	O
approaches	O
to	O
eliminating	O
some	O
variables	O
between	O
polynomials	O
of	O
several	O
variables	O
,	O
in	O
order	O
to	O
solve	O
systems	O
of	O
polynomial	O
equations	O
.	O
</s>
<s>
Classical	O
elimination	B-Algorithm
theory	I-Algorithm
culminated	O
with	O
the	O
work	O
of	O
Francis	O
Macaulay	O
on	O
multivariate	O
resultants	O
,	O
as	O
described	O
in	O
the	O
chapter	O
on	O
Elimination	B-Algorithm
theory	I-Algorithm
in	O
the	O
first	O
editions	O
(	O
1930	O
)	O
of	O
Bartel	O
van	O
der	O
Waerden	O
's	O
Moderne	O
Algebra	O
.	O
</s>
<s>
After	O
that	O
,	O
elimination	B-Algorithm
theory	I-Algorithm
was	O
ignored	O
by	O
most	O
algebraic	O
geometers	O
for	O
almost	O
thirty	O
years	O
,	O
until	O
the	O
introduction	O
of	O
new	O
methods	O
for	O
solving	O
polynomial	O
equations	O
,	O
such	O
as	O
Gröbner	O
bases	O
,	O
which	O
were	O
needed	O
for	O
computer	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
The	O
field	O
of	O
elimination	B-Algorithm
theory	I-Algorithm
was	O
motivated	O
by	O
the	O
need	O
of	O
methods	O
for	O
solving	O
systems	O
of	O
polynomial	O
equations	O
.	O
</s>
<s>
The	O
case	O
of	O
linear	O
equations	O
was	O
completely	O
solved	O
by	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
where	O
the	O
older	O
method	O
of	O
Cramer	O
's	O
rule	O
does	O
not	O
proceed	O
by	O
elimination	O
,	O
and	O
works	O
only	O
when	O
the	O
number	O
of	O
equations	O
equals	O
the	O
number	O
of	O
variables	O
.	O
</s>
<s>
Nevertheless	O
Hilbert	O
's	O
Nullstellensatz	O
,	O
may	O
be	O
considered	O
to	O
belong	O
to	O
elimination	B-Algorithm
theory	I-Algorithm
,	O
as	O
it	O
asserts	O
that	O
a	O
system	O
of	O
polynomial	O
equations	O
does	O
not	O
have	O
any	O
solution	O
if	O
and	O
only	O
if	O
one	O
may	O
eliminate	O
all	O
unknowns	O
to	O
obtain	O
the	O
constant	O
equation	O
1	O
=	O
0	O
.	O
</s>
<s>
Elimination	B-Algorithm
theory	I-Algorithm
culminated	O
with	O
the	O
work	O
of	O
Leopold	O
Kronecker	O
,	O
and	O
finally	O
Macaulay	O
,	O
who	O
introduced	O
multivariate	O
resultants	O
and	O
U-resultants	O
,	O
providing	O
complete	O
elimination	O
methods	O
for	O
systems	O
of	O
polynomial	O
equations	O
,	O
which	O
are	O
described	O
in	O
the	O
chapter	O
on	O
Elimination	B-Algorithm
theory	I-Algorithm
in	O
the	O
first	O
editions	O
(	O
1930	O
)	O
of	O
van	O
der	O
Waerden	O
's	O
Moderne	O
Algebra	O
.	O
</s>
<s>
Later	O
,	O
elimination	B-Algorithm
theory	I-Algorithm
was	O
considered	O
old-fashioned	O
and	O
removed	O
from	O
subsequent	O
editions	O
of	O
Moderne	O
Algebra	O
.	O
</s>
<s>
It	O
was	O
generally	O
ignored	O
until	O
the	O
introduction	O
of	O
computers	O
,	O
and	O
more	O
specifically	O
of	O
computer	B-Algorithm
algebra	I-Algorithm
,	O
which	O
again	O
made	O
relevant	O
the	O
design	O
of	O
efficient	O
elimination	O
algorithms	O
,	O
rather	O
than	O
merely	O
existence	O
and	O
structural	O
results	O
.	O
</s>
<s>
The	O
main	O
methods	O
for	O
this	O
renewal	O
of	O
elimination	B-Algorithm
theory	I-Algorithm
are	O
Gröbner	O
bases	O
and	O
cylindrical	O
algebraic	O
decomposition	O
,	O
introduced	O
around	O
1970	O
.	O
</s>
<s>
There	O
is	O
also	O
a	O
logical	O
facet	O
to	O
elimination	B-Algorithm
theory	I-Algorithm
,	O
as	O
seen	O
in	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
This	O
is	O
the	O
case	O
of	O
the	O
theory	O
of	O
polynomials	O
over	O
an	O
algebraically	O
closed	O
field	O
,	O
where	O
elimination	B-Algorithm
theory	I-Algorithm
may	O
be	O
viewed	O
as	O
the	O
theory	O
of	O
the	O
methods	O
to	O
make	O
quantifier	O
elimination	O
algorithmically	O
effective	O
.	O
</s>
