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+\documentclass[12pt,a4paper]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{tikz,tikz-cd}
+\usepackage{enumitem}
+\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
+
+\title{Mathematical software - homework 1}
+\author{Sebastiano Tronto}
+
+\newtheorem*{thm}{Theorem}
+\newtheorem{prop}{Proposition}
+
+\theoremstyle{definition}
+\newtheorem{ex}{Exercise}
+
+\theoremstyle{definition}
+\newtheorem{remark}{Remark}
+
+\begin{document}
+
+\noindent\hrulefill
+
+\begin{center}
+\Huge{\textbf{Mathematical Software - Retake}}
+\end{center}
+
+\noindent\hrulefill
+\begin{center}
+\begin{tabular}{lcr}
+\texttt{sebastiano.tronto@uni.lu} & \qquad \qquad \qquad \qquad &
+\textbf{Deadline: Wednesday, February 9th}
+\end{tabular}
+\end{center}
+
+\vspace{0.3cm}
+
+\begin{center}
+  \emph{\large For exercises 1 and 2 submit a .tex and a .pdf file.
+    For exercises 3 and 4 submit your Sage code either in text format (.txt or
+    .sage) or as a Jupyter Notebook file (.ipynb).
+  }
+\end{center}
+
+
+\section*{Exercise 1}
+  Write a short Latex document that contains the following theorem-like
+  environments using the \texttt{\textbackslash newtheorem} command of the
+  \texttt{amsthm} package (the box around the text is not needed):
+  \begin{center}
+    \fbox{\parbox{0.95\textwidth}{
+      \begin{prop}[Fundamental Theorem of Algebra]
+        \label{prop:fta}
+        Let \(p(x)\) be a non-constant polynomial with coefficients in
+        $\mathbb C$. Then there is \(z\in\mathbb C\) such that $p(z)=0$.
+      \end{prop}
+
+      \begin{remark}
+        Proposition \ref{prop:fta} is not true for polynomials with
+        coefficients in $\mathbb R$. For example
+        \begin{align}
+          p(x) = x^2+1
+        \end{align}
+        does not have real roots.
+      \end{remark}
+
+      \begin{thm}
+        If $X$ and $Y$ are $\sigma$-finite measure spaces and $f:X\times Y\to
+        \mathbb R$ is measurable and such that
+        \begin{align*}
+          \int_{X\times Y}|f(x,y)|\mathrm d(x,y) < \infty
+        \end{align*}
+        then
+        \begin{align}
+          \label{eq:fubini}
+          \int_X\left(\int_Yf(x,y)\mathrm d y\right)\mathrm d x =
+          \int_Y\left(\int_Xf(x,y)\mathrm d x\right)\mathrm d y =
+          \int_{X\times Y} f(x,y)\mathrm d(x,y)\,.
+        \end{align}
+      \end{thm}
+
+      \begin{remark}
+        In practice, equation \eqref{eq:fubini} means that we can switch the
+        order of integration in a double integral.
+      \end{remark}
+    }}
+  \end{center}
+  Notice that Propositions, Remarks and some of the equations are numbered,
+  and some of them are referred to in the Remarks. This numbering should change
+  accordingly if more numbered Theorems and equations are added before this
+  part of the text.
+
+\newpage
+
+\section*{Exercise 2}
+Create a Latex document containing the following pictures:
+\begin{enumerate}
+\item[(a)] The following commutative diagram:
+        \begin{center}
+                \begin{tikzcd}
+                        M \ar[r, "f"] \ar[d,swap,"i",hook] & A \\
+                        N \ar[ur, swap, "\tilde f", dashed]
+                \end{tikzcd}
+        \end{center}
+\item[(b)] A triangle with verteces on a grid, as below.
+           Moreover, the position of the vertex $C$ below must be easy to
+           change at will: you should use the
+           \texttt{\textbackslash pgfmathsetmacro} command to set a value
+           for its coordinates at the beginning, so that changing only 
+           those numbers makes the whole picture change accordingly (sides,
+           dots, labels).
+      \begin{center}
+        \begin{tikzpicture}[scale=1]
+                \pgfmathsetmacro{\ax}{2}
+                \pgfmathsetmacro{\ay}{1}
+                \pgfmathsetmacro{\bx}{8}
+                \pgfmathsetmacro{\by}{1}
+                \pgfmathsetmacro{\cx}{9}
+                \pgfmathsetmacro{\cy}{7}
+
+                \draw[lightgray!30,thin] (0,0) grid (10,10);
+                \draw[-] (\ax, \ay) -- (\bx, \by) -- (\cx, \cy) -- cycle;
+                \filldraw[blue] (\ax, \ay) circle[radius=0.1] node[below left] {$A$};
+                \filldraw[blue] (\bx, \by) circle[radius=0.1] node[below right] {$B$};
+                \filldraw[red] (\cx, \cy) circle[radius=0.1] node[above] {$C$};
+
+        \end{tikzpicture}
+      \end{center}
+ 
+\end{enumerate}
+
+\newpage
+\section*{Exercise 3}
+Use SageMath to solve the following problems:
+\begin{enumerate}[label=(\arabic*)]
+        \item Find the roots of the following polynomial over $\mathbb Q$,
+                over $\mathbb R$ and over $\mathbb C$:
+                \[ p(x) = x^6+x^5-2x^4-3x^3-x^2+2x+2 \]
+        \item Find the determinant, the trace and the characteristic polynomial
+                   of the following matrix:
+                \[A=
+                \left(\begin{array}{rrrrr}
+                2 & 3 & 0 & 1 & 2 \\
+                1 & 0 & \frac{1}{2} & 1 & -1 \\
+                0 & 0 & -1 & 0 & 0 \\
+                0 & -1 & -1 & 0 & -1 \\
+                -1 & -1 & -1 & -1 & 0
+                \end{array}\right)
+                \]
+        \item Find the solutions of the linear system $A\mathbf x=\mathbf 0$, where
+                $A$ is the matrix above and $\mathbf 0$ is the zero vector.
+        \item Find the points of intersection of the circle of equation $x^2+y^2=4$
+                   ad the ellipse of equation $\left(\frac{x}{2}\right)^2+(2y)^2=4$
+        \item Plot the graph of the function $f(x)=\sqrt{1-x^6}$.
+        \item Find the area between the $x$-axis and the grap of the function of
+                   the previous point.
+        \item Find the derivative, a primitive (integral) and the Taylor series
+              expansion up to order 4 of the function $h(x)=\log(1+x+x^2)$.
+        \item Use Sage to get the Latex code for the objects you computed in
+                the previous point.
+        \item Find a solution for the differential equation with initial conditions
+                   \[
+                        \begin{cases}g'(x)&=\frac{1}{3}g(x) - 7\\g(1)&=30\end{cases}
+                   \]
+        \item Draw a bar chart of the data set $[0,1,3,7,5,7,2,8,9,3]$, like the foll
+              following:
+                \begin{center}
+              \includegraphics[scale=0.5]{bc.png}
+              \end{center}
+\end{enumerate}
+
+\newpage
+\section*{Grading}
+
+\vspace{0.3cm}
+\textbf{Exercise 1 (5 points).}
+\begin{itemize}
+  \item A correct use of the \texttt{\textbackslash newtheorem} command is
+        worth 2 out of 5 points.
+  \item A correct use of the labelling and reference system is worth 2 points.
+  \item Reproducing correctly the mathematical formulas is worth 1 point.
+\end{itemize}
+
+\vspace{0.3cm}
+\textbf{Exercise 2 (5 points).}
+\begin{itemize}
+        \item Part (a) is worth 2 points: 1 point for having the letters
+              $M$, $N$ and $A$ in the correct position and the arrows
+              pointing between them and 1 point for the style of the arrows
+              and the labels $f$, $i$ and $\tilde f$ in the correct position.
+        \item Part (b) is worth 3 points: 1 point for drawing a triangle, 1
+              point for the other decorative elements (colored dots, grid lines
+              and labels) and 1 point for having the point $C$ correctly set
+              as a macro so that it can be changed easily.
+\end{itemize}
+
+\vspace{0.3cm}
+\textbf{Exercise 3 (10 points).}
+\begin{itemize}
+        \item Each of the 10 parts is worth 1 point.
+\end{itemize}
+
+
+
+\end{document}