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Let denote the set of
(noncommutative) polynomials in
the indeterminates

over a field *K*.
is a noncommutative algebra.
Let ,
*C* be a set of polynomials
and
with each
.
If *r* is a common zero of
, then *r* is a
zero of every polynomial *p* which lies in the smallest
ideal generated by *C*.
Furthermore,
if is a generating set for
the ideal ,
then *r* is a common zero of the *p*'s if
and only if it is a common zero of the *q*'s.
These ideas are learned by most mathematics undergraduates
in the case that the indeterminates lie in a commutative algebra
and they also hold when the lie in a noncommutative algebra.
The practical value of doing this algebraic manipulation is
that
instead of trying to solve for the common zeros
of directly, it could be advantageous
to find a different set
of polynomials
which generates the same ideal
as and *then * solve for
the common zeros on the *q*'s.
For example, might consist of
decoupled polynomial equations
which could be solved numerically (say using Matlab)
while the original set *C* produces numerically intractable problems.

**
The reader who is interested in seeing examples at this point
might decide to skip directly to
§.
**

*Helton *

Wed Jul 3 10:27:42 PDT 1996