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\begin{center} {\large \textbf{Math 437 -- Homework 12}} \smallskip \\ 
Due 10:15am on Thursday, April 20, 2017  \\ \ \\
 \end{center}

Please indicate any sources you used for a given problem on the solution to that problem. 
For example, if you worked with another student to get the solution to a problem, please indicate who.
You are welcome to work together in small groups, but please try the problems on your own first and write up 
your own solutions. \\ \medskip 




\begin{prob}
Consider the polynomial ring $\Q[q,d,n,p]$ in variables $q=$ quarters, $d=$ dimes,
 $n=$ nickels,  and $p =$ pennies, and the ideal
\[ I = \langle p^5 - n, p^{10} - d, p^{25} - q\rangle \ \ \subset \ \ \Q[q,d,n,p].\]
\begin{itemize} 
\item[(a)] Using computational software, compute a Grobner basis for $I$ with respect to $>_{lex}$ and $>_{grlex}$ with $q>d>n>p$. 
\item[(b)] By hand, compute the normal form of $dn^3p^{7}$ with respect to $I$ using $>_{lex}$ and $>_{grlex}$.
\item[(c)] Give an interpretation of the normal form of $q^{a_1}d^{a_2}n^{a_3}p^{a_4}$ with respect to $>_{lex}$ and $>_{grlex}$.
\end{itemize}

\end{prob}\ \\




\begin{prob} (CLO \S2.8, \#7) 
Let $S\subseteq \R^3$ be formed by taking the union of straight lines joining pairs of points $(t,0,1)$ and $(0,1,t)$. 
Then $S$ can be given in the parametric form $(x,y,z) = u(t,0,1) + (1-u)(0,1,t)$, giving 
\begin{align*}
x&=ut\\
y & = 1-u\\
z & = u + t - ut
\end{align*}
\begin{itemize}
\item[(a)] Find a polynomial $f(x,y,z)$ for which $V(f)$ contains $S$. 
\item[(b)] Show that $V(f) = S$ (that is, every point $(x,y,z)\in V(f)$ is obtained by some values of $t, u$).  Hint: Try to ``solve'' $f=0$
for one variable as a function of the other two. 
\end{itemize}



\end{prob}\ \\





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