deletions | additions       diff --git a/bits2.tex b/bits2.tex  index 67928e0..a8a938b 100644  --- a/bits2.tex  +++ b/bits2.tex 
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\\  In this section we will introduce these polynomials, together with their   operations. \\  A polynomial in the variable $x$ and coefficients in $\Fb$ (i.e. a polynomial   \emph{over the field of bits})  is an expression of the form $a_0+a_1x+a_2x^2+...+a_nx^n$, where $a_0,...,a_n \i \Fb$ are  bits.  \\  The set containing all the polynomials in the variable $x$, whose coefficients   are bits is denoted by $\Fb [x]$, meaning that we are defining polynomials   \emph{over the field of bits}: [x]$:  $$\Fb [x] := \{1,x,x+1,x^2,x^2+x,....\}.$$  The notation is the analogue of that for polynomials over $\RR$, whose set is   usually denoted by $\RR[x]$.\\  In analogy with $\RR[x]$, we can define the \emph{degree} of a monomial   $x^\alpha \in \Fb[x]$ as the \emph{positive integer number} $\alpha$. For   example, the degree of $x^5$ is actually $5$.\\  The degree of a polynomial $p$ is the maximal degree of the monomials appearing   in $p$ with nonzero coefficient. Since the monomials in $\Fb[x]$ are only power   of $x$, the degree of $p$ is actually the maximal power of $x$ appearing in $p$.   The degree of the polynomial $p=0$ is conventionally set to $-\infty$.  As an example, the degree of $x^5+x^4+x^2+x$ is again $5$.  We explain now how to sum two polynomials.   Actually, the operation is defined exactly as that with polynomials over   $\mathbb{R}$, but one has to take into account that the coefficients are bits,  
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As we explained for multiples, we can underline that also raising to powers is   an operation with peculiar properties, making this process different from the   analogous for polynomials over $\RR$.  \begin{Exercise}\label{degree}  Find the degree of the following polynomials  \begin{itemize}  \item $(x^4+x^3+1)\cdot (x^3+x^2+1) $  \item $(x^3+x^2+x)+(x^4+x^3)$  \item $(x^2+x+1)+(x^2+x)$  \end{itemize}  Give your comments on the degree of $p\cdot q$ and $p+q$, with $p,q \in \Fb[x]$   and prove them.  \end{Exercise}  Consider for example the polynomial $p(x)=x+1\in \Fb[x]$ and suppose to compute  
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\end{Exercise}  In analogy with $\RR[x]$, we can define the \emph{degree} of a monomial   $x^\alpha \in \Fb[x]$ as the \emph{positive integer number} $\alpha$. For   example, the degree of $x^5$ is actually $5$.\\  The degree of a polynomial $p$ is the maximal degree of the monomials appearing   in $p$ with nonzero coefficient. Since the monomials in $\Fb[x]$ are only power   of $x$, the degree of $p$ is actually the maximal power of $x$ appearing in $p$.   The degree of the polynomial $p=0$ is conventionally set to $-\infty$.  As an example, the degree of $x^5+x^4+x^2+x$ is again $5$.  \begin{Exercise}\label{degree}  Find the degree of the following polynomials  \begin{itemize}  \item $(x^4+x^3+1)\cdot (x^3+x^2+1) $  \item $(x^3+x^2+x)+(x^4+x^3)$  \item $(x^2+x+1)+(x^2+x)$  \end{itemize}  Give your comments on the degree of $p\cdot q$ and $p+q$, with $p,q \in \Fb[x]$   and prove them.  \end{Exercise}  Each polynomial $p \in \Fb[x]$ is associated to a function  $$\widetilde{p}: \Fb \rightarrow \Fb $$  called \emph{evaluation} and defined as follows