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\def\title{Homework 5, March 14}
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\nopagenumbers

\bs
These problems are due Wednesday, March 28, after break.

\bs

{\bf Problem 1:} Compute a Gr\"obner basis for the equations 
$$
\align
a^3&=bc
\\
b^3&=ac
\\
c^3&=ab
\endalign
$$
When we discussed this in class, we were thinking of an ordering for
monomials where higher degree monomials were always bigger, but now I want you
to use the lexicographic ordering on monomials.  Using this Gr\"obner basis,
produce a set of monomials, all of which are minimal elements of the Gr\"obner
graph, which serve an $F$-vector space basis for the quotient ring.

\ms

{\bf Problem 2:}  Compute a Gr\"obner basis for the equations 
$$
\align
x^3+y^3&=11
\\
x^2+y^2&=1
\endalign
$$
Can you say anything about the number of solutions of these equations over the
complex numbers?

\ms

{\bf Problem 3:} Let $F$ be a field, let $R= F[T_1,\dots,T_n]$, let $I =
R f_1 + \dots R f_t$ be an ideal of $R$, and let $\Cal F$ be the 
family of replacement rules corresponding to the generators $f_1, \dots, f_t$.
Show that the graph $\Gamma(\Cal F)$ is confluent provided the confluency
property holds for any pair of arrows originating in a monomial.

\bs
The following exam question is to be done independently.
\bs

{\bf Exam Problem 2:} Let $F$ be a field, and let $R= F[T_1,\dots,T_n]$.  We
consider a total ordering of the monomials in $R$ such that each variable $T_i$
is greater than $1$, and if $p$ and $q$ are monomials satisfying $p \le q$,
then $p T_i \le q T_i$.  Show that the monomials are well-ordered by this
ordering.

\bye


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