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diff --git a/section_Multivariate_polynomials_on_bits__.tex b/section_Multivariate_polynomials_on_bits__.tex
index 32c4528..b2fec30 100644
--- a/section_Multivariate_polynomials_on_bits__.tex
+++ b/section_Multivariate_polynomials_on_bits__.tex
...
\deg(x^iy^j) \,=\, i+j\,.
$$
If we consider $xy^5 \in \Fb[x,y]$,
we have
$deg_x(xy^5)=1$, $deg_y(xy^5)=5$, $\deg_x(xy^5)=1$, $\deg_y(xy^5)=5$,
and
$deg(xy^5)=6$. $\deg(xy^5)=6$. \\
The \emph{degree of a polynomial} $f\in \Fb[x,y] $ is the maximal degree of the monomials appearing
in $f$ with nonzero coefficient, so, if $f=x^3y+xy^6-y^2\in \Fb[x,y]$,
$deg(f)=7$ $\deg(f)=7$ and if $g=x^3+0y^{12}$ then
$deg(g)=3$. $\deg(g)=3$.
\begin{Exercise}\label{Gradi2var}
In $\Fb[x,y]$ what is...
\begin{itemize}
...
\item $x_1-\frac{1}{2}x_2$ is not a polynomial in $\Fb[x_1,x_2]$;
\item $x_1x_2x_3^3+1$ is a polynomial in $\Fb[x_1,x_2,x_3]$.
\end{itemize}
When (as we
will usually do) have done before) we will deal with two or three variables, we will denote them as $x,y$ or $x,y,z$, so for example $xy+1\in \Fb[x,y]$, $x^4y+z^3 \in \Fb[x,y,z]$.
\\
For $1 \leq j\leq n$, the \emph{$j$-degree} of a term in $n$ variables $x_1^{i_1}\cdots x_n^{i_n}$ is the value $i_j$. In formulas
$$deg_j(x_1^{i_1}\cdots $$\deg_j(x_1^{i_1}\cdots x_n^{i_n})=i_j.$$
The \emph{total degree} (or, simply, degree)
of a term in $n$ variables $x_1^{i_1}\cdots x_n^{i_n}$ is the sum of all
$deg_j$ $\deg_j$ for all $1 \leq j\leq n$, i.e.
$$deg(x_1^{i_1}\cdots $$\deg(x_1^{i_1}\cdots x_n^{i_n}) =\sum_{j=1}^n deg_j(x_1^{i_1}\cdots x_n^{i_n}) =i_1+...+i_n.$$
If we consider $x_1x_3^5 \in
\Fb[x_1,x_2,x_3]$, \Fb[x_1,x_2,x_3,x_4]$,
we have
$deg_1(x_1x_3^5)=1$, $deg_2(x_1x_3^5)=0$,
$deg_3(x_1x_3^5)=5$ $\deg_1(x_1x_3^5)=1$, $\deg_2(x_1x_3^5)=0$,
$\deg_3(x_1x_3^5)=5$, $\deg_4(x_1x_3^5)=0$ and
$deg(x_1x_3^5)=6$. $\deg(x_1x_3^5)=6$. \\
The \emph{degree of a polynomial} $f\in \Fb[x_1,...,x_n] $ is the maximal degree of the monomials appearing
in $f$ with nonzero coefficient, so, if $f=x^3y+xz^6-z^2\in \Fb[x,y,z]$,
$deg(f)=7$ $\deg(f)=7$ and if $g=x^3+0y^{12}$ then
$deg(g)=3$. $\deg(g)=3$.
\begin{Exercise}\label{Gradi}
In
$\Fb[x_1,x_2x_3]$ $\Fb[x_1,x_2x_3,x_4]$ what is...
\begin{itemize}
\item the $2$-degree of $x_2^3x_3$?
\item the
$4$-degree of $x_2^3x_3$?
\item the degree of
$x_2^3x_3+x_2$? $x_2^3x_3+x_2x_4$?
\item the degree of
$x_1^7+x_2^4x_3^3$? $x_1^7+x_2^4x_3^3+x_4^3x_1$?
\end{itemize}
\end{Exercise}
Two polynomials $f,g \in \Fb[x_1,...,x_n]$ are called \emph{equal} if
...
if and only if
$x_1^{i_1}\cdots x_n^{i_n}$ appears in $g$ with coefficient $a_{i_1...i_n}$ as well.}
\end{center}
Then, we can see that, in $\Fb[x,y]$,
$x^3+0y^{12}=x^3=x^3+0xy^{32}$. $x^3+0y^{12}=x^3=x^3+0xy^{32}$ and in $\Fb[x_1,x_2,x_3,x_4,x_5]$,
$x_5x_4-x_2=x_5x_4+0x_3-x_2$.
\\
As seen in section \ref{Sec:Polynomials} for univariate polynomials, the sum and the product of polynomials in $\Fb[x_1,...x_n]$ are defined as for multivariate polynomials over $\RR$, only taking into account that their coefficients are bits.
