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##

Some conventions and notation

Throughout this thesis, , , and will
be used to denote the integer, rational, and real rings respectively.
For the remainder of this section it will always be assumed that is
in prenex form (see [Mishra(1993)] p. 356 for a discussion of converting an
arbitrary formula to prenex form). Specifically, it will be assumed
that

where
and is a quantifier free
formula.
For any
, the * level* of is defined
as the largest such that
. If is a set of
polynomials, will denote the -level elements of .

Finally, the ``'' symbol will be used to denote evaluation, so if
is a point in free-variable space, means
evaluated at , as discussed in the previous section.

*Christopher W Brown*

*2001-07-10*