.gitignore

@@ -15,4 +15,7 @@ auto/
 template/
 *.old
 *.fmt
-local/
+local/
+*.acn
+*.glo
+*.ist

thesis/chapters/gb/algebraic_structures.tex

@@ -30,8 +30,9 @@ Let us first start with a definition of a simpler structure than a group:
   \item $\left( a \circ b \right) \circ c = a \circ \left( b \circ c \right)$ for all $a, b, c \in
     M$,
   \item there is $e \in M$ such that $e \circ a = a \circ e = a$ for all $a \in M$.
-  \end{enumerate} A monoid is called a \textbf{commutative monoid} if, in
-  addition to (i) and (ii), the following axiom also holds:
+  \end{enumerate}
+  A monoid is called a \textbf{commutative monoid} if, in addition to (i) and
+  (ii), the following axiom also holds:
   \begin{itemize}
   \item[(iii)] $a \circ b = b \circ a$ for all $a, b \in M$.
   \end{itemize}
@@ -54,10 +55,11 @@ operations associated with the structure.
 The element $b$ in the definition above is called the \textbf{inverse} of $a$.
 Note that Abelian groups are commutative groups.
 
-\begin{df} \label{ring_df} A \textbf{ring} is a set $R$ with two binary
-  operations $\left( a, b \right) \mapsto a + b$ and $\left( a, b \right) \mapsto a \cdot b$,
-  referred to as addition and multiplication, such that the following axioms
-  hold:
+\begin{df}
+  \label{ring_df}
+  A \textbf{ring} is a set $R$ with two binary operations $\left( a, b \right) \mapsto
+  a + b$ and $\left( a, b \right) \mapsto a \cdot b$, referred to as addition and
+  multiplication, such that the following axioms hold:
   \begin{enumerate}[label=(\roman*)]
   \item the set $R$ is an Abelian group under addition with the \textbf{additive
       identity} $0$,
@@ -86,8 +88,9 @@ write $ab$.
   \end{enumerate}
 \end{es}
 
-\begin{df} \label{ideal_df} Let $R$ be a ring and $\emptyset \ne I \subseteq R$. Then $I$ is an
-  \textbf{ideal} of $R$ if:
+\begin{df}
+  \label{ideal_df}
+  Let $R$ be a ring and $\emptyset \ne I \subseteq R$. Then $I$ is an \textbf{ideal} of $R$ if:
   \begin{enumerate}[label=(\roman*)]
   \item $a+b \in I$ for all $a, b \in I$, and
   \item $ar \in I$ for all $a \in I$ and $r \in R$.
@@ -105,7 +108,8 @@ under multiplication by any $r \in R$.
   \end{enumerate}
 \end{pr}
 
-\begin{p} ~\begin{enumerate}[label=(\roman*)]
+\begin{p}
+  ~\begin{enumerate}[label=(\roman*)]
   \item Suppose $a \in R$. Then
     \begin{eqnarray*} a + a \cdot 0 & = & a \cdot 1 + a \cdot 0 \\
                                 & = & a \left( 1 + 0 \right) \\
@@ -122,16 +126,19 @@ under multiplication by any $r \in R$.
   \end{enumerate}
 \end{p}
 
-There is an analogy from modular arithmetic that illustrates an intuitive view
-of ideals --- they can be regarded as a generalization of a zero in a number set
-such as the integers. Consider the ring $\mathbb{Z}_n$ of integers modulo a
-given integer $n \in \mathbb{Z}$. The exact set of integers that we identify with
-$0$ in $\mathbb{Z}_n$ is the set $n \mathbb{Z} = \set*{nm | m \in \mathbb{Z}}$.
-This set meets the criteria for being an ideal ((i) and (ii) of definition
-\ref{ideal_df}) of $\mathbb{Z}$ and its elements ``behave'' like $0$ in
-$\mathbb{Z}$: adding two elements of $n \mathbb{Z}$ yields another element of $n
-\mathbb{Z}$ and multiplying any element of $n \mathbb{Z}$ again yields an
-element of $n \mathbb{Z}$.
+\begin{re}
+  \label{ideal_re}
+  There is an analogy from modular arithmetic that illustrates an intuitive view
+  of ideals --- they can be regarded as a generalization of a zero in a number
+  set such as the integers. Consider the ring $\mathbb{Z}_n$ of integers modulo
+  a given integer $n \in \mathbb{Z}$. The exact set of integers that we identify
+  with $0$ in $\mathbb{Z}_n$ is the set $n \mathbb{Z} = \set*{nm | m \in
+    \mathbb{Z}}$. This set meets the criteria for being an ideal ((i) and (ii)
+  of definition \ref{ideal_df}) of $\mathbb{Z}$ and its elements ``behave'' like
+  $0$ in $\mathbb{Z}$: adding two elements of $n \mathbb{Z}$ yields another
+  element of $n \mathbb{Z}$ and multiplying any element of $n \mathbb{Z}$ again
+  yields an element of $n \mathbb{Z}$.
+\end{re}
 
 Considering our definition of rings, note that an ideal might not be a ring
 itself. For example, consider the ring of integers and its ideal consisting of
@@ -166,3 +173,33 @@ We will often work with the finite field $\mathbb{Z}_2$, which merits a short
 comment. We will denote this field $\mathbb{F}_2$. The additive and
 multiplicative identities are $0$ and $1$, respectively. The additive inverse of
 $0$ is $0$. The element $1$ is also its additive and multiplicative inverse.
+
+% \begin{df}
+%   Let $F$ be a field. An $\mathbf{F}$\textbf{-vector space} $V$ is an Abelian
+%   group with addition and an additional operation $\circ : F \times V \mapsto V$, called
+%   \textbf{scalar multiplication}, such that for all $\alpha, \beta \in F$ and $v, w \in V$,
+%   the following properties hold:
+%   \begin{enumerate}[label=(\roman*)]
+%   \item $\alpha \circ \left( v + w \right) = \alpha \circ v + \alpha \circ w$,
+%   \item $\left( \alpha + \beta \right) \circ v = \alpha \circ v + \beta \circ v$,
+%   \item $\left( \alpha \cdot \beta \right) \circ v = \alpha \circ \left( \beta \circ v \right)$, and
+%   \item $1 \circ v = v$.
+%   \end{enumerate}
+% \end{df}
+
+% We call the elements of the vector space $V$ \textbf{vectors}, whereas the
+% elements of the field $F$ are called \textbf{scalars}. Even though the zero
+% scalar and zero vector are different objects, we will denote them both by 0.
+
+% \begin{df}
+%   Let $v_1, \ldots, v_n$ be pairwise different vectors in an $F$-vector space. Then
+%   the sum \[\sum_{i = 1}^{n} \alpha_i \cdot v_i \quad \left( \alpha_i \in F \text{ for } 1 \le i \le n
+%     \right)\] is a \textbf{linear combination} of the $v_i$ with coefficients
+%   $\alpha_i$.
+% \end{df}
+
+% \begin{e}
+%   Let $F$ be a field and $1 \le n \in \mathbb{N}$. Define an addition on $F^n$ by
+%   setting \[\left( v_1, \ldots, v_n \right) + \left( w_1, \ldots, w_n \right) = \left(
+%       v_1 + w_1, \ldots, v_n + w_n\right)\] where $v_i, w_i \in F$ for $1 \le i \le n$.
+% \end{e}

thesis/chapters/gb/gb.tex

@@ -1,5 +1,20 @@
 \chapter{Gröbner Bases}
 
+The theory of Gröbner bases for ideals in polynomial rings was introduced by
+Bruno Buchberger \cite{bb}, who named the concept in honor of his advisor
+Wolfgang Gröbner (1899--1980). Buchberger also developed the fundamental
+algorithm for the computation a Gröbner basis. A similar concept for ideals in power
+series rings was also introduced by Heisuke Hironaka \cite{hh1}, \cite{hh2}.
+
+Gröbner bases are discussed in multiple books including \cite{gb} and
+\cite{iva}. We will follow these books along the way as we gradually unveil the
+elegance and power of Gröbner bases in solving systems of polynomial equations.
+Further information can be also found in \cite{igb} and \cite{gbs}.
+
 \input{chapters/gb/algebraic_structures}
 
-\input{chapters/gb/multivariate_polynomials}
+\input{chapters/gb/multivariate_polynomials}
+
+\input{chapters/gb/monomial_orderings}
+
+\input{chapters/gb/multivariate_division}

thesis/chapters/gb/monomial_orderings.tex

@@ -0,0 +1,302 @@
+\section{Monomial Orders}
+
+A Gröbner basis always pertains to a particular order on monomials. Let us
+therefore introduce the most fundamental ones.
+
+Before we actually define a monomial order, let us start with a concise
+discussion about binary relations so that it is convenient to prove that certain
+orders are in fact monomial orders.
+
+\begin{df}
+  Let $S$ be a non-empty set. A \textbf{binary relation} on $S$ is a subset $r$
+  of $S \times S$. The relation $\Delta\!\left( S \right) = \set*{\left( a, a \right)~|~a
+    \in S}$ is the \textbf{diagonal} of $S$.
+\end{df}
+
+We will use only binary relations in our work and so we will refer to them
+simply as relations. In order to simplify the notation, we will also employ
+infix notation to denote that two elements are in a relation, i.e., if $r$ is a
+binary relation on $S$ and $a, b \in S$, then $a~r~b$ will mean $\left( a, b
+\right) \in r$.
+
+\begin{df}
+  \sloppy Let $r$ and $s$ be relations on $S$. The relation $r^{-1} =
+  \set*{\left( a, b \right) | \left( b, a \right) \in r}$ is the \textbf{inverse}
+  of $r$. The \textbf{strict part} of $r$ is the relation $r_s = r \setminus r^{-1}$,
+  and \[s \circ r = \set*{\left( a, c \right)~|\text{ there is } b \in S \text{ such
+        that } \left( a, b \right) \in r \text{ and } \left( b, c \right) \in s}\]
+  is the \textbf{product} of $r$ and $s$.
+\end{df}
+
+\begin{df}
+  Let $r$ be a relation on $S$. Then $r$ is
+  \begin{enumerate}[label=(\roman*)]
+  % \item \textbf{reflexive} if $\Delta\!\left( S \right) \subseteq r$,
+  \item \textbf{transitive} if $r \circ r \subseteq r$,
+  \item \textbf{antisymmetric} if $r \cap r^{-1} \subseteq \Delta\!\left( S \right)$,
+  \item \textbf{connex} if $r \cup r^{-1} = S \times S$,
+  % \item a \textbf{partial order on} $S$ if $r$ is reflexive, transitive and
+  %   antisymmetric,
+  \item a \textbf{linear order on} $S$ if $r$ is transitive, antisymmetric and
+    connex.
+  \end{enumerate}
+\end{df}
+
+\begin{df}
+  Let $r$ be a relation on $S$ with strict part $r_s$ and let $R \subseteq S$. An
+  element $a \in R$ is \textbf{minimal} if there is no $b \in R$ such that
+  $b~r_s~a$. A \textbf{strictly descending} (or \textbf{strictly decreasing})
+  \textbf{sequence} in $S$ is an infinite sequence of elements $a_n \in S$ such
+  that $a_{n+1}~r_s~a_n$ for all $n \in \mathbb{N}$. The relation $r$ is
+  \textbf{noetherian} if every non-empty subset $R$ of $S$ has a minimal
+  element. The relation $r$ is a \textbf{well-order} on $S$ if it is a
+  noetherian linear order on $S$.
+\end{df}
+
+A natural way to think about the strict part of a relation is to consider the
+natural order on $\mathbb{N}$, which is a linear order, where for each $m, n \in
+\mathbb{N}$; $m > n$ means $m \ge n$ and $m \ne n$. The symbol $>$ denotes the
+strict part of the relation $\ge$. We will also denote our orders on monomials by
+$\succeq$, the inverse will be $\preceq$ and the strict parts will be denoted
+$\succ$ and $\prec$.
+
+We will denote by $\mathcal{M}\!\left( x_1, \ldots, x_n \right)$, or simply
+$\mathcal{M}$, the set of all monomials in the indeterminates $x_1, \ldots, x_n$. It
+turns out that $\mathcal{M}$ forms an Abelian monoid under natural
+multiplication where we add corresponding exponents of the indeterminates. The
+multiplicative identity is the monomial 1. Note that we can associate any
+monomial $x^\alpha \in \mathcal{M}\!\left( x_1, \ldots, x_n \right)$ with its $n$-tuple of
+exponents $\alpha = \left(\alpha_1, \ldots, \alpha_n\right) \in \mathbb{N}_0^n$ in a one-to-one
+fashion. Thus, we can use the sets $\mathcal{M}$ and $\mathbb{N}_0^n$
+interchangeably.
+
+\begin{lm}
+  \label{order_lm}
+  A linear order $\ge$ on $S$ is a well-order if and only if there is no
+  strictly descending sequence in $S$.
+\end{lm}
+
+\begin{p}
+  Let us turn the lemma into its contrapositive form: $\ge$ is not a well-order if
+  and only if there is a strictly descending sequence in $S$; and prove this
+  version of the lemma.
+
+  \begin{itemize}
+  \item[$\implies$] Suppose $\ge$ is not a well-order. Then there is a non-empty
+    subset $R \subseteq S$ that has no minimal element. We can choose $a \in R$ and
+    since $a$ is not the minimal element, we can choose again $b \in R$ such that
+    $a > b$, which leads to a strictly descending sequence.
+
+  \item[$\impliedby$] Suppose there is a strictly descending sequence in $S$.
+    The elements of such a sequence form a non-empty subset $R$ of $S$ that has
+    no minimal element. Hence, $\ge$ is not a well-order. \qedhere
+  \end{itemize}
+\end{p}
+
+\begin{df}
+  \label{monomial_order_df}
+  A \textbf{monomial order} $\succeq$ is a well-order on $\mathcal{M}$, which
+  satisfies the \textbf{property of respecting multiplication}: if $m_1 \succeq m_2$,
+  then $n \cdot m_1 \succeq n \cdot m_2$ for all $m_1, m_2, n \in \mathcal{M}$.
+\end{df}
+
+The purpose of the property of respecting multiplication is that the relative
+ordering of monomials in a polynomial does not change when we multiply the
+polynomial by a monomial. Such behavior is necessary for the division algorithm
+described in the next section.
+
+\begin{df}[Lexicographic order]
+  Let $x^\alpha, x^\beta \in \mathcal{M}\!\left( x_1, \ldots, x_n \right)$ be monomials. We say
+  $x^\alpha \succeq_{lex} x^\beta$ if $\alpha = \beta$ or if there is $1 \le i \le n$ such that $\alpha_j = \beta_j$
+  for $1 \le j < i$ and $\alpha_i > \beta_i$.
+\end{df}
+
+Note that $\succ_{lex}$ compares the exponent $n$-tuples $\alpha, \beta \in \mathbb{N}_0^n$
+so that $x^\alpha \succ_{lex} x^\beta$ if the left-most non-zero component of the difference
+$\alpha - \beta \in \mathbb{N}_0^n$ is positive.
+
+\begin{re}
+  \label{lex_re}
+  Also note that the lexicographic order depends on how the underlying
+  indeterminates $x_1, x_2, \ldots, x_n$ are ordered. In general, there are $n!$ ways
+  to order $n$ indeterminates and each of these orders has its respective
+  lexicographic order. We will only assume the standard order where $x_1 > x_2 >
+  \cdots > x_n$, or the alphabetical order where $x > y > z$.
+\end{re}
+
+\begin{e}
+  \label{lex_e}
+  ~\begin{enumerate}[label=(\roman*)]
+  \item Let $xy^2z^3$ and $xy^3$ be monomials in $\mathcal{M}\!\left( x, y, z
+    \right)$. Then $xy^3 \succ_{lex} xy^2z^3$ since there is $i = 2$ and $j = 1$
+    such that $\alpha_j = \beta_j$ and $\alpha_i > \beta_i$, where $\alpha = (1, 3, 0)$ and $\beta = (1, 2,
+    3)$. Also, the left-most non-zero component of the difference $\beta - \alpha =
+    \left( 0, 1, -3 \right)$ is positive.
+  \item Let $x, y, z$ be monomials in $\mathcal{M}\!\left( x, y, z \right)$.
+    Then considering remark \ref{lex_re} and example (i), we get $x \succ_{lex} y
+    \succ_{lex} z$.
+  \item In the lexicographic order, note that a monomial that contains the most
+    significant indeterminate (as regards the underlying order) is greater than
+    any other monomial that does not contain such an indeterminate. For example,
+    if $x$ and $y^3z^2$ are monomials in $\mathcal{M}\!\left( x, y, z \right)$,
+    then $x \succ_{lex} y^3z^2$. The reasoning is the same as in (i) and (ii).
+  \end{enumerate}
+\end{e}
+
+The intuitive outlook on the lexicographic order is that it looks for the most
+significant indeterminate that appears in one of the monomials and then gives
+preference to the monomial in which this indeterminate has greater power.
+
+\begin{pr}
+  \label{lex_pr}
+  The lexicographic order $\succeq_{lex}$ on $\mathcal{M}$ is a monomial order.
+\end{pr}
+
+\begin{p}
+  Following the definition of the lexicographic order and the fact that the
+  regular numerical order on $\mathbb{N}_0$ is a linear order, it is
+  straightforward to show that for any monomials $x^{\alpha}, x^\beta, x^\gamma \in
+  \mathcal{M}\!\left( x_1, \ldots, x_n \right)$ and $\alpha, \beta, \gamma \in \mathbb{N}_0^n$,
+  the following conditions hold:
+  \begin{description}
+  \item[(transitivity)] if $x^\alpha \succeq_{lex} x^\beta$ and $x^\beta \succeq_{lex} x^\gamma$, then $x^\alpha
+    \succeq_{lex} x^\gamma$;
+  \item[(antisymmetry)] if $x^\alpha \succeq_{lex} x^\beta$ and $x^\alpha \preceq_{lex} x^\beta$, then
+    $x^\alpha = x^\beta$; and
+  \item[(connexity)] either $x^\alpha \succeq_{lex} x^\beta$ or $x^\alpha \preceq_{lex} x^\beta$.
+  \end{description}
+  These properties show that $\succeq_{lex}$ is a linear order on $\mathcal{M}$.
+
+  Let us prove the property of respecting multiplication explicitly. If $x^\alpha
+  \succeq_{lex} x^\beta$, then either $\alpha = \beta$, or there is $1 \le i \le n$ such that $\alpha_i -
+  \beta_i > 0$ with $\alpha_j = \beta_j$ for $1 \le j < i$. Also, $x^\alpha \cdot x^\gamma = x^{\alpha + \gamma}$ and
+  $x^\beta \cdot x^\gamma = x^{\beta + \gamma}$. Comparing the results gives us $\left( \alpha + \gamma \right)
+  - \left( \beta + \gamma \right) = \alpha - \beta$ and we see that $\alpha_i - \beta_i > 0$ with $\alpha_j =
+  \beta_j$ for $1 \le j < i$ again; or if $\alpha = \beta$, then $\left( \alpha + \gamma \right) = \left(
+    \beta + \gamma \right)$. This shows that also $x^{\alpha+\gamma} \succeq_{lex} x^{\beta+\gamma}$.
+
+  The last part to prove is to show that $\succeq_{lex}$ is also noetherian, i.e a
+  well-order. We will prove this by the following contradiction:
+
+  By lemma \ref{order_lm}, if $\succeq_{lex}$ is not a well-order, then there is a
+  strictly decreasing sequence \[\mathlarger{x^{\alpha_{(1)}} \succ_{lex} x^{\alpha_{(2)}}
+      \succ_{lex} \cdots}\] of elements in $\mathcal{M}\!\left( x_1, \ldots, x_n \right)$,
+  where each $\alpha_{(i)} = \left( \alpha_1, \ldots, \alpha_n \right) \in \mathbb{N}_0^n$. By the
+  definition of $\succeq_{lex}$, we also know that there exists a $j$ such that all
+  the first components of the $n$-tuples $\alpha_{(k)}$ with $k \ge j$ are equal.
+  Continuing further, there is an $l \ge j$ such that all the second components of
+  the $n$-tuples $\alpha_{(m)}$ with $m \ge l$ are all equal. We see that there must be
+  a $p \ge l$, for which the whole $n$-tuples $\alpha_{(p)} = \alpha_{(p+1)} = \cdots$ are all
+  equal. This means that the sequence is not strictly decreasing, which
+  contradicts the lemma.
+\end{p}
+
+\begin{df}[Reverse Colexicographic Order]
+  \sloppy Let $x^\alpha, x^\beta \in \mathcal{M}\!\left( x_1, \ldots, x_n \right)$ be monomials.
+  We say $x^\alpha \succeq_{rclex} x^\beta$ if $\alpha = \beta$ or if there is $1 \le i \le n$ such that
+  $\alpha_j = \beta_j$ for $i < j \le n$ and $\alpha_i < \beta_i$.
+\end{df}
+
+Observe that $\succ_{rclex}$ compares the exponent $n$-tuples $\alpha, \beta \in
+\mathbb{N}_0^n$ so that $x^\alpha \succ_{rclex} x^\beta$ if the right-most non-zero component
+of the difference $\alpha - \beta \in \mathbb{N}_0^n$ is negative. Remark \ref{lex_re} also
+applies.
+
+\begin{e}
+  \label{rclex_e}
+  ~\begin{enumerate}[label=(\roman*)]
+  \item Let $xy^2z^3$ and $xy^3$ be monomials in $\mathcal{M}\!\left( x, y, z
+    \right)$. Then $xy^3 \succ_{rclex} xy^2z^3$ as well as in example \ref{lex_e}
+    (i), but for a different reason. There is $i = 3$ such that $\alpha_i < \beta_i$,
+    where $\alpha = (1, 3, 0)$ and $\beta = (1, 2, 3)$. Also, the right-most non-zero
+    component of the difference $\beta - \alpha = \left( 0, 1, -3 \right)$ is negative.
+  \item The lexicographic order coincides with the reverse colexicographic order
+    for monomials in one and two indeterminates. These orders may differ for
+    monomials in three and more variables, as shown by the following example:
+    let $xz$ and $y^2$ be monomials in $\mathcal{M}\!\left( x, y, z \right)$.
+    Then $xz \succ_{lex} y^2$, as explained in example \ref{lex_e} (i), but $y^2
+    \succ_{rclex} xz$, as explained in example (i).
+  \end{enumerate}
+\end{e}
+
+The intuitive outlook on the reverse colexicographic order is that it looks for
+the least significant indeterminate that appears in one of the monomials and
+then gives preference to the monomial in which this indeterminate has lesser
+power. It can be thought of as a double reversal of the lexicographic order ---
+we first reverse the underlying order of the indeterminates and then their
+powers.
+
+Equivalently to the lexicographic order, it is straightforward to show that the
+reverse colexicographic order is a linear order as well. However, it is not a
+well-order since it is possible to define the following strictly decreasing
+sequence \[x_1x_2 \succ_{rclex} x_1x_2^2 \succ_{rclex} x_1x_2^3 \succ_{rclex} \cdots\] of
+monomials in $\mathcal{M}\!\left( x_1, x_2 \right)$. In this sequence, let $x^\alpha
+= x^{(1,n)}$ and $x^\beta = x^{(1,n+1)}$ for $n \in \mathbb{N}_{>0}$. We see that it
+is always the case that $x^\alpha \succ_{rclex} x^\beta$ since $\alpha_1 = \beta_1$ and $\alpha_2 < \beta_2$,
+and we get a strictly decreasing sequence. Hence, by lemma \ref{order_lm},
+$\succeq_{rclex}$ is not a well-order and by definition \ref{monomial_order_df},
+$\succeq_{rclex}$ cannot be a monomial order either. For this reason, we will not use
+it to order monomials on its own, but we will use it as a ``sub-order'' in the
+definition of the next order, which will be a monomial order.
+
+Examples \ref{lex_e} and \ref{rclex_e} show that the lexicographic and reverse
+colexicographic orders do not take into consideration the total degree of
+monomials. Later in our work, we will see that in certain cases, it is desirable
+to order the monomials in a polynomial according to their total degree. Let us
+therefore introduce the following order, which allows for the total degree.
+
+\begin{df}[Graded Reverse Lexicographic Order]
+  \sloppy Let $x^\alpha, x^\beta \in \mathcal{M}\!\left( x_1, \ldots, x_n \right)$ be monomials.
+  We say $x^\alpha \succeq_{grlex} x^\beta$ if $\lvert x^\alpha \rvert > \lvert x^\beta \rvert$, or
+  $\lvert x^\alpha \rvert = \lvert x^\beta \rvert$ and $x^\alpha \succeq_{rclex} x^\beta$.
+\end{df}
+
+Notice that despite its name, the graded reverse lexicographic order actually
+makes use of the reverse colexicographic order. There is a general consensus on
+such a name, so we will follow it.
+
+\begin{e}
+  ~\begin{enumerate}[label=(\roman*)]
+  \item Let $x, y^2, xz \in \mathcal{M}\!\left( x, y \right)$ be monomials. Then
+    $y^2 \succeq_{grlex} x$ since $\lvert y^2 \rvert = 2 > \lvert x \rvert = 1$; and
+    $y^2 \succeq_{grlex} xz$ since $\lvert xz \rvert = \lvert y^2 \rvert$ and $y^2
+    \succeq_{rclex} xz$.
+  \item Let $x, y, z \in \mathcal{M}\!\left( x, y z \right)$ be monomials. Then $x
+    \succeq_{grlex} y \succeq_{grlex} z$ since $\lvert x \rvert = \lvert y \rvert = \lvert z
+    \rvert$ and $x \succeq_{rclex} y \succeq_{rclex} z$.
+  \end{enumerate}
+\end{e}
+
+\begin{pr}
+  The graded reverse lexicographic order $\succeq_{grlex}$ on $\mathcal{M}$ is a
+  monomial order.
+\end{pr}
+
+\begin{p}
+  Since $\succeq_{grlex}$ first uses the usual well-order order on the total degree of
+  monomials $\lvert x^\alpha \rvert \in \mathbb{N}_0$ and when $\lvert x^\alpha \rvert =
+  \lvert x^\beta \rvert$, it decides ties using the reverse colexicographic order
+  (which is a linear order), grlex is also linear.
+
+  It is also straightforward to show that $\succeq_{grlex}$ is a well-order since we
+  consider only the strict part $\succ_{grlex}$, which is solely the well-order on
+  $\lvert x^\alpha \rvert \in \mathbb{N}_0$.
+
+  In order to show that the property of respecting multiplication holds,
+  consider the monomials $x^\alpha, x^\beta, x^\gamma \in \mathcal{M}\!\left( x_1, \ldots, x_n
+  \right)$ with the $n$-tuples $\alpha, \beta, \gamma \in \mathbb{N}_0^n$. Also, $x^\alpha \cdot x^\gamma =
+  x^{\alpha+\gamma}$ and $x^\beta \cdot x^\gamma = x^{\alpha+\gamma}$. Assume $x^\alpha \succeq_{grles} x^\beta$. If $\lvert x^\alpha
+  \rvert > \lvert x^\beta \rvert$, then $x^{\alpha+\gamma} \succ_{grlex} x^{\beta+\gamma}$ since $\lvert
+  x^{\alpha+\gamma} \rvert = \lvert x^\alpha \rvert + \lvert x^\gamma \rvert > \lvert x^\beta \rvert +
+  \lvert x^\gamma \rvert = \lvert x^{\beta+\gamma} \rvert$. Also, if $\lvert x^\alpha \rvert =
+  \lvert x^\beta \rvert$, we get $\lvert x^{\alpha+\gamma} \rvert = \lvert x^{\beta+\gamma} \rvert$ by
+  the same argument as above and we use the reverse colexicographic order. So if
+  $\lvert x^\alpha\rvert = \lvert x^\beta \rvert$, then $x^\alpha \succeq_{rclex} x^\beta$ (since we
+  have assumed that $x^\alpha \succeq_{grlex} x^\beta$\big), which means that either $\alpha = \beta$,
+  or there is $1 \le i \le n$ such that $\alpha_i - \beta_i < 0$ with $\alpha_j = \beta_j$ for $i < j
+  \le n$. As in the proof of proposition \ref{lex_pr}, comparing the results gives
+  us $\left( \alpha + \gamma \right) - \left( \beta + \gamma \right) = \alpha - \beta$ and we see that $\alpha_i
+  - \beta_i < 0$ with $\alpha_j = \beta_j$ for $i < j \le n$ again; or if $\alpha = \beta$, then $\left(
+    \alpha + \gamma \right) = \left( \beta + \gamma \right)$. This shows that $x^{\alpha+\gamma} \succeq_{grlex}
+  x^{\beta+\gamma}$ and completes the proof.
+\end{p}

thesis/chapters/gb/multivariate_polynomials.tex

@@ -1,5 +1,203 @@
 \section{Multivariate Polynomials}
 
+% Polynomials are the moving spirit behind Gröbner bases.
+
+\begin{df}
+  Let $\alpha = \left( \alpha_1, \ldots, \alpha_n \right)$ be an $n$-tuple of non-negative integers.
+  A \textbf{monomial} in $x_1, \ldots, x_n$ is a product of the form \[\prod_{i=1}^n
+    x_i^{\alpha_i} = x_1^{\alpha_1} \cdot x_2^{\alpha_2} \cdots x_n^{\alpha_n}.\]
+\end{df}
+
+Let us simplify the notation by setting \[x^\alpha = \prod_{i=1}^n x_i^{\alpha_i}.\] The
+\textbf{total degree} of a monomial $x^\alpha$ is the sum $\sum_{i=1}^n \alpha_i.$ We
+simplify the notation again an let $\lvert x^\alpha \rvert$ denote the total degree
+of $x^\alpha$. We will call the symbols $x_1, \ldots, x_n$ indeterminates or variables,
+depending on which context we will need to emphasize. Note that $x^\alpha = 1$ when
+$\alpha = \left( 0, \ldots, 0 \right)$ and also when $\lvert x^\alpha \rvert = 0$. Also note
+that any monomial is fully determined by $\alpha$.
+
+\begin{df}
+  Let $x^\alpha$ be a monomial and let $K$ be a field. A \textbf{term} with a
+  non-zero \textbf{coefficient} $c_\alpha \in K$ is the product $c_\alpha x^\alpha$.
+\end{df}
+
+\begin{df}
+  A \textbf{polynomial} $f$ with coefficients in a field $K$ is a finite sum of
+  terms in the form \[f = \sum_\alpha c_\alpha \cdot x^\alpha, \quad c_\alpha \in K.\] The zero polynomial will
+  be denoted $0$.
+\end{df}
+
+\begin{df}
+  Let $f = \sum c_\alpha x^\alpha \ne 0$ be a non-zero polynomial. The \textbf{total
+    degree} of $f$, denoted deg$\left( f \right)$, is the maximum $\lvert x^\alpha
+  \rvert$ such that the corresponding coefficient $c_\alpha$ is nonzero. The degree
+  of $0$ is undefined.
+\end{df}
+
+The set of all polynomials in $x_1, \ldots, x_n$ with coefficients in a field $K$
+will be denoted $K\!\left[ x_1, \ldots, x_n \right]$. When the particular
+indeterminates are of no relevance, we will denote the set by
+$K\!\left[\mathbf{x} \right]$ for short. We will also employ the standard
+letters $x, y$ and $z$ instead of $x_1, x_2$ and $ x_3$ when we discuss
+illustrative polynomials.
+
+Let $f,g \in K\!\left[ \mathbf{x} \right]$ be polynomials. We say that $f$
+\emph{divides} $g$ if $g = fh$ for some polynomial $h \in K\!\left[ \mathbf{x}
+\right]$. One can show that the set $K\!\left[ \mathbf{x} \right]$ satisfies all
+of the ring axioms under standard polynomial addition and multiplication. We
+will therefore refer to $K\!\left[ \mathbf{x} \right]$ as a \emph{polynomial
+  ring}. Not all polynomials in this ring have their multiplicative inverses,
+e.g., even the elementary polynomial $x_1$ does not have its multiplicative
+inverse and so $K\!\left[ \mathbf{x} \right]$ does not form a field. A proof
+that $K\!\left[ \mathbf{x} \right]$ forms a ring can be found in
+\cite[Chapher~2]{gb}, the authors also provide a broader outlook on polynomials
+by defining them in a more abstract way.
+
 \begin{df}
-  A \textbf{monomial} in $x$
-\end{df}
+  Let $\set*{f_1, \ldots, f_s} \subset K\!\left[ \mathbf{x} \right]$ be a set of
+  polynomials. Then we set \[\langle f_1, \ldots, f_s \rangle = \set*{\sum_{i=1}^s
+      h_if_i~\bigg|~h_1, \ldots, h_s \in K\!\left[ \mathbf{x} \right]}.\]
+\end{df}
+
+\begin{lm}
+  If $\set*{f_1, \ldots, f_s} \subset K\!\left[ \mathbf{x} \right]$ is a set of
+  polynomials, then $\langle f_1, \ldots, f_s \rangle$ is an ideal of $K\!\left[ \mathbf{x}
+  \right]$.
+\end{lm}
+
+\begin{p}
+  Assume $f = \sum_{i=1}^s p_if_i$ and $g = \sum_{i=1}^s q_if_i$ are polynomials, and
+  let also $h \in K\!\left[ \mathbf{x} \right]$ be a polynomial. Then the
+  equations
+  \begin{align*}
+    f + g &= \sum_{i=1}^s \left( p_i + q_i \right) f_i \text{\quad and} \\
+       hf &= \sum_{i=1}^s \left( hp_i \right) f_i
+  \end{align*}
+  show that $\langle f_1, \ldots, f_s \rangle$ meets the criteria for being an ideal of
+  $K\!\left[ \mathbf{x} \right]$.
+\end{p}
+
+\begin{df}
+  Let $\set*{f_1, \ldots, f_s} \subset K\!\left[ \mathbf{x} \right]$ be a set of
+  polynomials and let $I$ be an ideal such that $I = \langle {f_1, \ldots, f_s} \rangle$. The set
+  $\set*{f_1, \ldots, f_s}$ is a \textbf{basis} of $I$. We will also call $\langle {f_1, \ldots,
+    f_s} \rangle$ the \textbf{ideal generated by} $\set*{f_1, \ldots, f_s}$.
+\end{df}
+
+Remark \ref{ideal_re} provides an intuitive view of ideals through modular
+arithmetic. Another analogy comes from linear algebra where the definition of
+subspaces can be likened to the definition of ideals of polynomial rings. Both
+are closed under addition. Subspaces are closed under multiplication by scalars
+while ideals of polynomial rings are closed under multiplication by polynomials.
+An ideal generated by a set of polynomials also shares similar properties with a
+span generated by a set of vectors, which is a structure similar to subspaces as
+well.
+
+\begin{df}
+  Let $K$ be a field and $n$ a positive integer. The $n$-dimensional
+  \textbf{affine space} over $K$ is the set \[K^n = \set*{\left( a_1, \ldots, a_n
+      \right)\ |\ a_1, \ldots, a_n \in K}.\] 
+\end{df}
+
+\begin{re} \label{poly_re}
+  A polynomial $f \in K\!\left[ x_1, \ldots, x_n \right]$ can be regarded as a function
+  $f : K^n \mapsto K$ that takes in points in the affine space $K^n$ and produces
+  elements of the field $K$.
+\end{re}
+
+\begin{df}
+  Let $K^n$ be an affine space and let $f = f\!\left( x_1, \ldots, x_n \right) \in
+  K\!\left[ x_1, \ldots, x_n \right]$ be a polynomial. The \textbf{zero point} of $f$
+  is a point $\left( a_1, \ldots, a_n \right) \in K^n$ such that $f\!\left( a_1, \ldots, a_n
+  \right) = 0$.
+\end{df}
+
+\begin{df}
+  Let $\set*{f_1, \ldots, f_s} \subset K\!\left[ x_1, \ldots, x_n \right]$ be a set of
+  polynomials and $K^n$ an affine space. The \textbf{affine variety} $V\!\left(
+    f_1, \ldots, f_s \right)$ defined by $\set*{f_1, \ldots, f_s}$ is the set \[V\!\left(
+      f_1, \ldots, f_s \right) = \set*{\left( a_1, \ldots, a_n \right) \in
+      K^n~\Big|~f_i\!\left( a_1, \ldots, a_n \right) = 0 \text{ for all } 1 \le i \le
+      s}\] of all zero points of all the polynomials in $\set*{f_1, \ldots, f_s}$.
+\end{df}
+
+Solving an equation that can be expressed as a polynomial in multiple variables
+can be seen as finding the zero points of the corresponding polynomial. Affine
+varieties generalize this notion to systems of polynomial equations. Considering
+remark \ref{poly_re}, we may also see varieties as geometric objects, which is briefly
+illustrated by the following example:
+
+\begin{e}
+  Consider the real coordinate space $\mathbb{R}^2$ and the polynomial $f =
+  f\!\left( x^2 + y^2 - 1 \right)$. The variety $V\!\left( f \right)$ is the
+  unit circle centered at the origin.
+\end{e}
+
+We will use the following lemma to show that a given ideal is contained in
+another one. This is useful for proving the equality of two ideals in example
+\ref{ideal_e}.
+
+\begin{lm}
+  \label{ideal_lm}
+  Let $I \subseteq K\!\left[ \mathbf{x} \right]$ be an ideal, and let $\set*{f_1, \ldots,
+    f_s} \subset K\!\left[ \mathbf{x} \right]$ be a set of polynomials. Then $\langle f_1,
+  \ldots, f_s \rangle \subseteq I$ if and only if $\set*{f_1, \ldots, f_s} \subseteq I$.
+\end{lm}
+
+\begin{ps}
+  \begin{itemize}
+  \item[$\implies$] Assume $\langle {f_1, \ldots, f_s} \rangle \subseteq I$. Each $f_i \in \set*{f_1, \ldots,
+      f_s}$ can be constructed as follows: $f_i = 0 \cdot f_1 + \cdots + 1 \cdot f_i + \cdots + 0
+    \cdot f_s$, and hence $\set*{f_1, \ldots, f_n} \subseteq I$.
+  \item[$\impliedby$] Assume $\set*{f_1, \ldots, f_n} \subseteq I$ and choose any $f \in \langle
+    {f_1, \ldots, f_s} \rangle$ so that $f = h_1f_1 + \cdots + h_sf_s$ where each $h_i \in
+    K\!\left[ \mathbf{x} \right]$. We see that $f \in I$ since $I$ is an
+    ideal and so $\langle {f_1, \ldots, f_s} \rangle \subseteq I$. \qedhere
+  \end{itemize}
+\end{ps}
+
+\begin{e}
+  \label{ideal_e}
+  Consider the ideals $\langle {x, y} \rangle$ and $\langle {x + y, x - y} \rangle$ in the polynomial
+  ring $\mathbb{Q}\!\left[ x, y \right]$. We will show that these two ideals are
+  equal so that $\langle {x, y} \rangle = \langle {x + y, x - y} \rangle$.
+
+  We see that $x + y \in \langle {x, y} \rangle$ and $x - y \in \langle {x, y} \rangle$, so by lemma
+  \ref{ideal_lm}, $\langle {x + y, x - y} \rangle \subseteq \langle {x, y} \rangle$. Similarly, both $x =
+  \frac{1}{2}\!\left( x + y \right) + \frac{1}{2}\!\left( x - y \right)$ and
+  $y = \frac{1}{2}\!\left( x + y \right) - \frac{1}{2}\!\left( x - y \right)$
+  are in $\langle {x + y, x - y} \rangle$ so that by lemma \ref{ideal_lm}, $\langle {x, y} \rangle \subseteq \langle
+  {x + y, x - y} \rangle$ and the equality follows.
+\end{e}
+
+\begin{pr}
+  \label{ideal_pr}
+  If $\set*{f_1, \ldots, f_s}$ and $\set*{g_1, \ldots, g_t}$ are two bases of the same
+  ideal in $K\!\left[ x_1, \ldots, x_n \right]$, so that $\langle f_1, \ldots, f_s \rangle =
+  \langle g_1, \ldots, g_t \rangle$, then $V\!\left( f_1, \ldots, f_s \right) = V\!\left(g_1, \ldots, g_t
+  \right)$. 
+\end{pr}
+
+\begin{p}
+  \sloppy Choose any $\left( a_1, \ldots, a_n \right) \in V\!\left( f_1, \ldots, f_s
+  \right)$. We know that all polynomials in $\set*{f_1, \ldots, f_s}$ are equal to
+  zero at $\left( a_1, \ldots, a_n \right)$. Now choose any $g \in \langle {g_1, \ldots, g_t} \rangle$.
+  Since $\langle {g_1, \ldots, g_t} \rangle = \langle {f_1, \ldots, f_s} \rangle$, we can write $g = \sum_{i=1}^s h_i
+  f_i$, $h_i \in K\!\left[x_1, \ldots, x_n \right]$. Then $g\!\left( a_1, \ldots, a_n
+  \right) = \sum_{i=1}^s h_i\!\left( a_1, \ldots, a_n \right) \cdot f_i\!\left( a_1, \ldots, a_n
+  \right) = 0,$ which shows that $\left( a_1, \ldots, a_n \right) \in V\!\left( g_1, \ldots,
+    g_t \right)$, which means that $V\!\left( f_1, \ldots, f_s \right) \subseteq V\!\left(
+    g_1, \ldots, g_t \right)$. The opposite inclusion can be proved in the same way.
+\end{p}
+
+Example \ref{ideal_e} shows that an ideal may have multiple different bases
+while proposition \ref{ideal_pr} reveals that a variety is actually determined
+by the ideal generated by its basis and not by the basis itself.
+
+A system of multivariate equations can be seen as an ideal basis. Proposition
+\ref{ideal_pr} then gives us a potential ability to change the original system
+to another one while keeping the exact same solution set. We will model our
+cipher as system of polynomial equations and then we will transform this system
+into a new one which will be solvable in linear time. We will show that a
+Gröbner basis is the new system and that the transformation will be the most
+demanding part of the computation as regards both time and memory.

thesis/ref.bib

@@ -1,39 +1,72 @@
-@MANUAL{svgspec,
-    title = "Scalable Vector Graphics (SVG) 1.1 Specification {[online]}",
-    organization = "WWW Consorcium",
-    url = "http://www.w3.org/TR/2003/REC-SVG11-20030114/",
-    note = "[cit. 2011-07-07]",
-    key = "svg"
-}
+@Book{gb,
+ author = {Becker, Thomas},
+ title = {Gröbner bases : a computational approach to commutative algebra},
+ publisher = {Springer-Verlag},
+ year = {1993},
+ address = {New York},
+ isbn = {0-387-97971-9}
+ } 
 
-@INBOOK{kobltypo,
-	title = "Praktick{\' a} typografie",
-	author = "Pavel Ko{\v c}i{\v c}ka and Filip Bla{\v z}ek",
-	publisher = "Brno: Computer Press",
-	year = "2004"
+@article{bb,
+  title={Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal},
+  author={Buchberger, Bruno},
+  journal={Journal of symbolic computation},
+  volume={41},
+  number={3-4},
+  pages={475--511},
+  year={2006},
+  publisher={Elsevier}
 }
 
-@BOOK{rybicka,
-	title = "LaTeX pro za\v c\'ate\v cn\'\i{}ky",
-	author = "Ji{\v r}{\' i} Rybi{\v c}ka",
-	isbn = "80-7302-049-1",
-	publisher = "Brno: Konvoj",
-	edition = "3"
+@Book{iva,
+ author = {Cox, David},
+ title = {Ideals, varieties, and algorithms : an introduction to computational algebraic geometry and commutative algebra},
+ publisher = {Springer},
+ year = {2015},
+ address = {Cham},
+ isbn = {9783319167206}
+ }
+
+@article{hh1,
+ ISSN = {0003486X},
+ URL = {http://www.jstor.org/stable/1970486},
+ author = {Heisuke Hironaka},
+ journal = {Annals of Mathematics},
+ number = {1},
+ pages = {109--203},
+ publisher = {Annals of Mathematics},
+ title = {Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I},
+ volume = {79},
+ year = {1964}
 }
 
-@MANUAL{iso690,
-	organization = "\'U\v rad pro technickou normalizaci, metrologii a st\'atn\'\i{} zku\v sebnictv\'\i{}",
-	title = "\v CSN ISO 690 Informace a dokumentace -- Pravidla pro bibliografick\'e odkazy a citace informa\v cn\'\i{}ch zdroj{\r u}",
-	year = "2011"
+@article{hh2,
+ ISSN = {0003486X},
+ URL = {http://www.jstor.org/stable/1970547},
+ author = {Heisuke Hironaka},
+ journal = {Annals of Mathematics},
+ number = {2},
+ pages = {205--326},
+ publisher = {Annals of Mathematics},
+ title = {Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II},
+ volume = {79},
+ year = {1964}
 }
 
+@Book{igb,
+ author = {Adams, William},
+ title = {An introduction to Gröbner bases},
+ publisher = {American Mathematical Society},
+ year = {1994},
+ address = {Providence, R.I},
+ isbn = {978-0-8218-3804-4}
+ }
 
-@ARTICLE {sitova-karta-s-fpga-xilinx,
-    title  = "Síťová karta s FPGA Xilinx pro 1 a 10GbE",
-    journal = "V: Pandatron - elektrotechnický magazín {[online]}",
-    howpublished = "[online]",
-    month  = "červen",
-    year   = "2013",
-    note = "[cit. 2015-10-31]",
-    url    = "http://pandatron.cz/?2939&sitova_karta_s_fpga_xilinx_pro_1_a_10gbe"
-}
+@Book{gbs,
+ author = {Hibi, Takayuki},
+ title = {Gröbner bases : statistics and software systems},
+ publisher = {Springer},
+ year = {2013},
+ address = {Tokyo New York},
+ isbn = {978-4-431-54573-6}
+ }

thesis/thesis.tex

@@ -8,29 +8,49 @@
 \documentclass[thesis=M,english]{FITthesis}[2019/12/23]
 
 \usepackage{dirtree} %directory tree visualisation
-
 \usepackage{mathtools}
 \usepackage{amsthm}
 \usepackage{amssymb}
 \usepackage{enumitem}
+\usepackage{relsize}
+
+\hypersetup{
+  colorlinks   = true, %Colours links instead of ugly boxes
+  urlcolor     = blue, %Colour for external hyperlinks
+  linkcolor    = blue, %Colour of internal links
+  citecolor   = red %Colour of citations
+}
 
-\newtheorem{thm}{Theorem}[subsection]
+\newtheorem{thm}[subsection]{Theorem}
 \newtheorem{pr}[subsection]{Proposition}
 \newtheorem{lm}[subsection]{Lemma}
 
-\theoremstyle{definition}
+\newtheoremstyle{dfstyle}% % Name
+  {}%                      % Space above
+  {}%                      % Space below
+  {}%                      % Body font
+  {}%                      % Indent amount
+  {\bfseries}%             % Theorem head font
+  {.}%                     % Punctuation after theorem head
+  { }%                     % Space after theorem head, ' ', or \newline
+  {\thmname{#1}\thmnumber{ #2}\thmnote{ (#3)}}
+
+\theoremstyle{dfstyle}
 \newtheorem{df}[subsection]{Definition}
+
+\theoremstyle{definition}
 \newtheorem{e}[subsection]{Example}
+\newtheorem{re}[subsection]{Remark}
 
 \newtheoremstyle{break}% name
-  {}%         Space above, empty = `usual value'
-  {}%         Space below
+  {}% Space above, empty = `usual value'
+  {}% Space below
   {}% Body font
-  {}%         Indent amount (empty = no indent, \parindent = para indent)
+  {}% Indent amount (empty = no indent, \parindent = para indent)
   {\bfseries}% Thm head font
-  {.}%        Punctuation after thm head
+  {.}% Punctuation after thm head
   {\newline}% Space after thm head: \newline = linebreak
-  {}%         Thm head spec
+  {}%  Thm head spec
 
 \theoremstyle{break}
 \newtheorem{examples}[subsection]{Example}
@@ -43,12 +63,16 @@
   \proof[\normalfont{\bfseries #1}]%
 }{\endproof}
 
+\newenvironment{ps}[1][\proofname]{%
+  \proof[\normalfont{\bfseries #1}]\leavevmode\vspace{-\topskip}%
+}{\endproof}
+
 \DeclarePairedDelimiter\set\{\}
 
 % % list of acronyms
-% \usepackage[acronym,nonumberlist,toc,numberedsection=autolabel]{glossaries}
-% \iflanguage{czech}{\renewcommand*{\acronymname}{Seznam pou{\v z}it{\' y}ch zkratek}}{}
-% \makeglossaries
+\usepackage[acronym,nonumberlist,toc,numberedsection=autolabel]{glossaries}
+\iflanguage{czech}{\renewcommand*{\acronymname}{Seznam pou{\v z}it{\' y}ch zkratek}}{}
+\makeglossaries
 
 \department{Department of Theoretical Computer Science}
 
@@ -105,9 +129,10 @@
 \appendix
 
 \chapter{Abbreviations and Symbols}
-% \printglossaries
+\printglossaries
 \begin{description}
-	\item[$\mathbb{N} = $] the set of natural numbers
+	\item[$\mathbb{N}_0 = $] the set of natural numbers including zero
+	\item[$\mathbb{N}_{>0} = $] the set of natural numbers excluding zero
   \item[$\mathbb{Z} = $] the set of integers
   \item[$\mathbb{Q} = $] the set of rational numbers (fractions)
   \item[$\mathbb{R} = $] the set of real numbers