NOTE: This section of the documentation is out of date!
In the Projection Phase, QEPCAD computes a set of polynomials
that implicitly defines the CAD that it will later explicitly
construct in the Stack Construction Phase. This set of
polynomials, the projection factor set, must satisfy
some special properties in order to define a CAD. So the goal
of this phase is to construct as small a set as we can while
ensuring that it defines a CAD and contains certain polynomials
we'll need to solve the quantifier elimination problem.
A projection operator is a function that's used to
define a projection factor set, and several are available in
QEPCAD.
The Average User
======================================================= Enter an informal description between '[' and ']': [ Using Hong's Projection at level 4 ] Enter a variable list: (a,b,d,c) Enter the number of free variables: 4 Enter a prenex formula: [ 54 c^3 - 27 a b c^2 + 6 a^3 c^2 - 144 b d c + 6 a^2 d c + 4 b^3 c - a^2 b^2 c + 96 a d^2 + 40 a b^2 d - 9 a^3 b d = 0 ]. ======================================================= Before Normalization > go Before Projection (c) > proj-op (m,m,h) Before Projection (c) > go Before Choice > go Before Solution >
Error! Delineating polynomial should be added over cell(2,2)!"
during the stack construction phase. In this case it turns out
that McCallum's projection is not invalid, it's just that QEPCAD
can't prove that it's valid. Thus we must rerun the
example with Hong's projection for all but the last two
projection steps. In general, if McCallum's projection produces
an error message for a problem involving variables
(x1,x2,...,xk) you should issue the command
proj-op (m,m,h,h,h,...,h) where the list has
k-1 elements, all but the first two of which are 'h'.
The Advanced User
If you know some things about different projection operators you
can interact with QEPCAD to do better - specifically by removing
projection factors during the projection process. I'll add some
info on that in the future!