deletions | additions       diff --git a/section_Multivariate_polynomials_on_bits__.tex b/section_Multivariate_polynomials_on_bits__.tex  index ae5f7c2..5eadc08 100644  --- a/section_Multivariate_polynomials_on_bits__.tex  +++ b/section_Multivariate_polynomials_on_bits__.tex 
...
We begin with two variables, $x$ and $y$.\\  A \emph{term} or \emph{monomial} in the $2$ variables $x$ and $y$ is a product of powers, i.e.   $x^{i} y^j$, y^h$,  for some $i,j$ $i,h$  in $\NN$.\\ For example, we can consider the following monomials $$x^2y^3 \,(i=2,j=3),\quad \,(i=2,h=3),\quad  x^4\, (i=4,j=0),\quad (i=4,h=0),\quad  y^7\, (i=0,j=7), (i=0,h=7),  \quad 1\, (i=j=0).$$ (i=h=0).$$  Be careful that for example $x^{-4}$ is \textbf{not} a monomial.  \\  With monomials, we can define polynomials in the $2$ variables $x$ and $y$ and coefficients in $\Fb$ (i.e. multivariate polynomials in the field of bits) as sums of monomials (without coefficients). For example, the following are polynomials 
...
Formally, a polynomial in the $2$ variables $x$ and $y$ and coefficients in $\Fb$ is any   expression of the form  $$  f(x,y)=\sum_{i,j f(x,y)=\sum_{i,h  \in \NN} a_{i,j} a_{i,h}  x^{i} {y}^{\,j} {y}^{h}  $$   such that  \begin{itemize}  \item for each $i,j$ $i,h$  in $\NN$, $ a_{i,j} a_{i,h}  \in \Fb$, \item only a \textbf{finite} number of coefficients is nonzero.   \end{itemize}  The set containing all polynomials in $x,y$ with bits as coefficient is denoted by $\Fb[x,y]$.\\  We can notice that the definition of multivariate polynomial over the field of bits is the analogous of that over $\RR$.  \\  The $x$-degree of a term $x^iy^j$ $x^iy^h$  in the two variables $x,y$ is the value $i$, whereas $j$ $h$  is its $y$-degree. In formulas   $$  \deg_x(x^{i}y^j)=i,\;\deg_y(x^{i}y^j)=j \deg_x(x^{i}y^h)=i,\;\deg_y(x^{i}y^h)=h  $$  The \emph{total degree} (or, simply, degree)   of a term $x^iy^j$ $x^iy^h$  in the two variables $x,y$ is the sum of the $x$-degree and the $y$-degree of  the monomial i.e.  $$  \deg(x^iy^j) \deg(x^iy^h)  \,=\, i+j\,. i+h\,.  $$  If we consider $xy^5 \in \Fb[x,y]$,   we have $\deg_x(xy^5)=1$, $\deg_y(xy^5)=5$, 
...
\\  As seen in section \ref{Sec:Polynomials} for univariate polynomials, the sum and the product of polynomials in $\Fb[x,y]$ are defined as for multivariate polynomials over $\RR$, only taking into account that their coefficients are bits.  \begin{Example}\label{SumEProdF2Bivar}  In $\Fb[x,y]$, let $f=x^3y+xy+1 $f_1=x^3y+xy+1  $, $g=y^3+xy+1 $f_2=y^3+xy+1  $ and $h=xy+1$. $f_3=xy+1$.  It holds \begin{itemize}  \item $f+g=(x^3y+xy+1)+(y^3+xy+1)=x^3y+y^3 $f_1+f_2=(x^3y+xy+1)+(y^3+xy+1)=x^3y+y^3  $ \item $hf=(xy+1)(x^3y+xy+1)=x^4y^2+x^2y^2+x^3y+1$ $f_3f_1=(xy+1)(x^3y+xy+1)=x^4y^2+x^2y^2+x^3y+1$  \end{itemize}  \end{Example}  \begin{Exercise}\label{operazionibivar} 
...
It is totally analogous to that in $2$ variables.  \\  A \emph{term} or \emph{monomial} in the $3$ variables$x$, $y$ and $z$ is a product of powers, i.e.   $x^{i} y^jz^l$, y^hz^l$,  for some $i,j,l$ $i,h,l$  in $\NN$.\\ For example, we can consider the following monomials $x^2y^3z$ ($i=2,j=3, ($i=2,h=3,  l=1$), $x^4$ ($i=4,j=l=0$), ($i=4,h=l=0$),  $y^7$ ($i=l=0,j=7$), ($i=l=0,h=7$),  $z^9$ ($i=j=0,l=9$), ($i=h=0,l=9$),  $1$ ($i=j=l=0$). ($i=h=l=0$).  Be careful that for example $z^{-6}$ is \textbf{not} a monomial.  \\  With monomials, we can define polynomials in the variables $x$, $y$ and $z$ and coefficients in $\Fb$ as sums of monomials (without coefficients). For example, $x^2y^3+x+x^10+z$, $xy+z^11y^2$, $xyz+z^2+1$ and $x^3y^5z$ (a monomial is also a polynomial). \\  Formally, a polynomial in the $3$ variables $x$, $y$ and $z$ and coefficients in $\Fb$ is any   expression of the form  $$  f(x,y,z)=\sum_{i,j,l f(x,y,z)=\sum_{i,h,l  \in \NN} a_{i,j,l} a_{i,h,l}  x^{i} y^{j}z^l y^{h}z^l  $$   such that  \begin{itemize}  \item for each $i,j,l$ $i,h,l$  in $\NN$, $ a_{i,j,l} a_{i,h,l}  \in \Fb$, \item only a \textbf{finite} number of coefficients is nonzero.   \end{itemize}  The set containing all polynomials in $x,y,z$ with bits as coefficient is denoted by $\Fb[x,y,z]$.\\  \\  The $x$-degree of a term $x^iy^jz^l$ $x^iy^hz^l$  in the two variables $x,y$ is the value $i$, whereas $j$ $h$  is its $y$-degree and $l$ its $z$-degree. In formulas   $$  \deg_x(x^{i}y^jz^l)=i,\;\deg_y(x^{i}y^jz^l)=j,\; deg_z(x^{i}y^jz^l)=l \deg_x(x^{i}y^hz^l)=i,\;\deg_y(x^{i}y^hz^l)=h,\; deg_z(x^{i}y^hz^l)=l  $$  The \emph{total degree} (or, simply, degree)   of a term $x^iy^jz^l$ $x^iy^hz^l$  in the three variables $x,y,z$ is the sum of the $x$-degree, the $y$-degree and the $z$-degree of  the monomial i.e.  $$  \deg(x^iy^jz^l) \deg(x^iy^hz^l)  \,=\, i+j+l\,. i+h+l\,.  $$  If we consider $xy^5z^3 \in \Fb[x,y,z]$,   we have $deg_x(xy^5z^3)=1$, $deg_y(xy^5z^3)=5$, 
...
\end{Exercise}  Two polynomials $f,g \in \Fb[x,y,z]$ are called \emph{equal} if   \begin{center}  \emph{a term $x_1^iy^jz^l$ $x_1^iy^hz^l$  appears in $f$ with nonzero coefficient $a_{i,j,l}$ $a_{i,h,l}$  \\ if and only if   $x_1^iy^jz^l$ $x_1^iy^hz^l$  appears in $g$ with coefficient $a_{i,j,l}$ $a_{i,h,l}$  as well.} \end{center}  Then, we can see that, in $\Fb[x,y,z]$, $x^3+0y^{12}=x^3=x^3+0xyz^{32}$ and $xyz=0x^3+xyz+0y^4+0z$.  \\