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diff --git a/section_Multivariate_polynomials_on_bits__.tex b/section_Multivariate_polynomials_on_bits__.tex
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+++ b/section_Multivariate_polynomials_on_bits__.tex
...
\ \\
\underline{We consider now the case of three variables, $x$, $y$ and
$z$.}
\\ $z$.}\\
%
A \emph{term} or \emph{monomial} in the three variables $x$, $y$ and $z$ is a product of powers, i.e.
$x^{i} y^hz^l$, for some $i,h,l$ in $\NN$.\\
For example, we can consider the following monomials $$x^2y^3z\, (i=2,h=3, l=1),\quad x^4\, (i=4,h=l=0),\quad z^9\, (i=h=0,l=9), \quad 1\, (i=h=l=0).$$
...
as raising to the power 2 its monomials and summing them.
\\
\underline{Now we are ready to tackle the generic case of $n$
variables.} variables: $x_1,x_2,...,x_n$}\\
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A \emph{term} or \emph{monomial} in the $n$ variables
$x_1,...x_n$ $x_1,...,x_n$ is a product of powers of
$x_1,...x_n$, $x_1,...,x_n$, i.e.
$x_1^{i_1}\cdots x_n^{i_n}$ for some $i_1,...,i_n \in \NN$.\\
For example, for three variables $x_1,x_2,x_3$, it is clear that $x_1^3x_2^8x_3^6$, $x_2x_3^4=x_1^0x_2x_3^4$ ,$x_1^4=x_1^4x_2^0x_3^0$ are all terms.
\\
...
\end{itemize}
The set containing all polynomials in $n$ variables $x_1,...x_n$ with coefficients in
$\Fb$ is denoted by $\Fb[x_1,...,x_n]$.\\
We can notice that the definition of multivariate polynomial over the field of bits is the analogous of that over $\RR$.
\\
From the definition of polynomial with $n$ variables and coefficients in a field, it holds that
\begin{itemize}
\item $x_2^2x_3+x_1$ is a polynomial in $\Fb[x_1,x_2,x_3]$; but it is \textbf{not} a polynomial in $\Fb[x_1,x_2]$;
\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 have done before) we will deal with two or three variables, we
will may denote them as $x,y$ or
$x,y,z$, so for example $xy+1\in \Fb[x,y]$, $x^4y+z^3 \in \Fb[x,y,z]$. $x,y,z$.
\\
For $1 \leq j\leq n$, the
\emph{$j$-degree} \emph{$x_j$-degree} of a term in $n$ variables $x_1^{i_1}\cdots x_n^{i_n}$ is the value $i_j$.
In formulas
$$\deg_j(x_1^{i_1}\cdots To simplify the expression we can use $\deg_j=\deg_{x_j}$ i.e.
$$\deg_{x_j}(x_1^{i_1}\cdots x_n^{i_n})=\deg_{j}(x_1^{i_1}\cdots x_n^{i_n})=i_j.$$
The \emph{total degree} (or, simply, degree)
of a term
$t$ in $n$ variables
$x_1^{i_1}\cdots $t=x_1^{i_1}\cdots x_n^{i_n}$ is the sum of all
$\deg_j$ $\deg_j(t)'$s for all $1 \leq j\leq n$, i.e.
$$\deg(x_1^{i_1}\cdots x_n^{i_n}) =\sum_{j=1}^n
deg_j(x_1^{i_1}\cdots \deg_j(x_1^{i_1}\cdots x_n^{i_n}) =i_1+...+i_n.$$
If we consider $x_1x_3^5 \in \Fb[x_1,x_2,x_3,x_4]$,
then $\deg_1(x_1x_3^5)=1$, $\deg_2(x_1x_3^5)=0$,
$\deg_3(x_1x_3^5)=5$, $\deg_4(x_1x_3^5)=0$ and $\deg(x_1x_3^5)=6$. \\
