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Jennifer and Andrew,
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Both your presentation and the final project paper were excellent.  You 
did a really good job making the $F_4$ algorithm understandable for
the class.  Your paper also presents everything very well.  I have a 
few relatively minor comments for you:
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\item{1.}  You say that ``Buchberger's Algorithm is the most standard
method of computing Gr\"obner bases in most computer algebra systems
such as Maple.''  While that was true in the past, the current 
``state of the art''
is actually much more interesting.  In fact, as you describe at the 
end of the paper, the $F_4$ algorithm itself is a big part of Maple's
Gr\"obner basis routine in the latest versions.  There is a discussion 
on pages 589-590 of all the different Gr\"obner basis algorithms that
are incorporated into the {\tt Basis} command that we used all the time
and there are actually even two different implementations of $F_4$
that are a part of that.  Maple selects an approach to use based
on the number of variables, how complicated the polynomials are, what
monomial order is specified, and so forth.  This is all transparent
to the user, but as you saw, it is possible to set the {\it infolevel}
option to see what's going on ``under the hood'' if you want to.   
A number of the different components of the Gr\"obner {\tt Basis} 
command in Maple
 are things we did not talk about, but that are
connected to some of the ideas we were seeing in the last lab and 
problem set (the FGLM procedure in particular).  
\item{2.}  A writing point:  It's OK to use first person in mathematical writing, 
especially if you want to ``bring the reader into'' what you 
are saying.  That way you can also avoid passive
constructions like the one you have here.  It sounds much
better to say ``we will outline the basic form of the $F_4$
algorithm'' than ``outlined will be the basic form of the $F_4$ algorithm.''
\item{3.}  It would have been much better to put the pseudocode first, and 
then explain what's going on.  I find this sort of written out form
of the algorithm is not easy to follow without having the pseudocode 
to look at.  The same goes for the later discussion of the {\tt ComputeM}
procedure.
\item{4.}  What you are saying about $F_4$ reducing to Buchberger if 
$B$ consists of a single pair is essentially true.  But I think there
are actually cases where elements of the
ideal corresponding to rows of the reduced echelon form matrix
would turn out to be different from the polynomials produced by 
the division algorithm computation 
of $\overline{S(f_i,f_j)}^G$.  The reason is that the row-reduction
of the matrix produced by {\tt ComputeM} is not restricted in the 
same way that choice of divisors in the division algorithm is restricted.
\item{5.}  Give reference to Theorem 2 in Section 10.3 of IVA.  
\item{6.}  The {\tt LinearAlgebra} package in Maple also has a 
row-reduced echelon form command called {\tt ReducedRowEchelonForm}.
Is there a reason you didn't want to use that?  The {\tt MTM} package
is really designed to let you combine Maple and MATLAB.
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Final Project Presentation:  98 (A)
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\noindent
Final Project Paper 96 (A)
\bye
