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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due October 19 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Fall 2016 $|$ M.~Macauley}}


\begin{document}

%$\;$

\noindent \emph{Read}: Chapter 3.1--3.4 of Robeva/Hodge:
\emph{Inferring the topology of gene regulatory networks: an algebraic
  approach to reverse engineering}. By B.~Stigler and E.~Dimitrova,
pages 75--90. \medskip

\noindent \emph{Do}:
\begin{enumerate}

%%---------------------------------------------------------------------

\item How many Boolean networks $f=(f_1,f_2,f_3)$ fit the following data?
  \[
  (1,1,1)\stackrel{f}{\longto}(1,1,0)\stackrel{f}{\longto}(0,0,1)\stackrel{f}{\longto}(0,0,1)\,.
  \]
  By inspection, find two of them. Express your answer using
  Boolean logic and as polynomials in $\F_2[x_1,x_2,x_3]$. Bonus points if one or both of your solutions were found by no one else in the class. \\

%%---------------------------------------------------------------------  

\item Consider the following \emph{time series} in a $3$-node
  polynomial dynamical system over $\F_3$:
  \[
  (1,1,1)\stackrel{f}{\longto}(2,0,1)\stackrel{f}{\longto}(2,0,0)\stackrel{f}{\longto}(0,2,2)\stackrel{f}{\longto}(0,2,2)\,.
  \]
For reference, here are the input vectors $\s_i$ and output vectors $\t_i$:
\begin{align*}
  \s_1&=(s_{11},s_{12},s_{13})=(1,1,1)\,,\qquad\t_1=(t_{11},t_{12},t_{13})=(2,0,1)\,,
  \\ 
  \s_2&=(s_{21},s_{22},s_{23})=(2,0,1)\,,\qquad\t_2=(t_{21},t_{22},t_{23})=(2,0,0)\,,
  \\
  \s_3&=(s_{31},s_{32},s_{33})=(2,0,0)\,,\qquad\t_3=(t_{31},t_{32},t_{33})=(0,2,2)\,, 
  \\
  \s_4&=(s_{41},s_{42},s_{43})=(0,2,2)\,,\qquad\t_4=(t_{41},t_{42},t_{43})=(0,2,2)\,.
\end{align*}
\begin{enumerate}
\item Find polynomials $f_1,f_2,f_3$ in $\F_3[x_1,x_2,x_3]$ that
  fit the data. That is, $f_j(\s_i)=\t_i$ for all $i=1,2,3,4$.
\item For each $j=1,2,3,4$, write down the ideal $I_j=I(\s_j)$ of
  polynomials that vanish on the data point $\s_j$.
\item Use the following commands in Macaulay2 to compute the ideal $I$
  of polynomials that vanish on \emph{all} of the input data points.
\begin{verbatim}
   R = ZZ/3[x1,x2,x3,MonomialOrder=>Lex];
   I = intersect{I1, I2, I3, I4};  
\end{verbatim}
Compute a Gr\"obner basis $\mathcal{G}$ of $I$.
\item Write the \emph{model space} of the time series using your answer to Part~(a) as the particular solution. 
\item Compute the normal form of $f_1,f_2,f_3$ with respect to $\mathcal{G}$ by reducing them modulo the ideal $I$. Write the model space using this particular solution.
\item Repeat Parts~(c)--(e) using \verb+MonomialOrder=>GRevLex+. \\

Turn in a print-out of your Macaulay2 worksheet.

\end{enumerate}


%In this exercise, we will find a polynomial dynamical systems
%  $f=(f_1,f_2,f_3)$ over $\Z_2$ that fits the following data:
%  \[
%  (1,1,0)\stackrel{f}{\longto}(0,0,1)\stackrel{f}{\longto}(0,0,1)\,.
%  \]


%%---------------------------------------------------------------------  


  % Robeva/Hodge, Exercise 2.8

%\newpage
%
%$\;$\vspace{.1in}





  
\end{enumerate}


\end{document}


