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\begin{center} {\large \textbf{Math 437 -- Homework 11}} \smallskip \\ 
Due 10:15am on Thursday, April 13, 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} (Graded Reverse Lexicographic Order) 
For $\alpha \neq \beta \in (\Z_{\geq0})^n$, say $x^{\alpha} >_{grevlex} x^{\beta}$ if
\[ \sum_{i=1}^n \alpha_i >\sum_{i=1}^n \beta_i, \ \ \text{ or } \ \
\sum_{i=1}^n \alpha_i =\sum_{i=1}^n \beta_i \ \text{ and the rightmost nonzero entry of $\alpha - \beta$ is negative}.
  \] \smallskip
\begin{itemize}
\item[(a)] Show that $>_{grevlex}$ is a monomial ordering. (\emph{Hint:} See Corollary 6 in \S2.4 of CLO)
\item[(b)] For $f = 5x_1^3+4x_2^4 +3x_1^2x_3^2 +2x_1+x_3$, find $LT(f)$ under each of $>_{lex}$, $>_{grlex}$, and $>_{grevlex}$. 
\end{itemize}
\end{prob}\ \\




\begin{prob} (CLO \S2.3 \#6) 
Using graded lexicographic order $ >_{grlex}$, 
find an element $g$ of $\langle f_1, f_2\rangle = \langle 2xy^2-x, 3x^2y - y-1\rangle \subset \Q[x,y]$ 
 whose remainder upon division by $\{f_1, f_2\}$ is non-zero.  Is $f_1, f_2$ a Gr\"obner basis for $\langle f_1, f_2\rangle$?  Explain. 
\end{prob}\ \\






\vfill 
\noindent \textbf{Some notes on the final project}\\


\noindent {\bf Length:} Your final project should be roughly 3-5 pages in length, with 1-1.5 line spacing.  It's fine if the paper is longer than 5 pages, but not
too much longer (10 pages is a hard maximum).  \\

\noindent {\bf Audience:} Think of your audience for this paper as other students in the class, people who have general mathematical background and familiarity with 
the course material but haven't read in-depth papers on your subject. \\


\noindent {\bf Rough draft:} If you would like me to read through a rough draft and give you comments, please give me a rough draft by Thursday, April 20.  (This is optional!)\\

\noindent {\bf Deadline:} The paper is due May 4 at 11am.  You can submit either a hard copy or pdf. 


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