deletions | additions       diff --git a/section_Multivariate_polynomials_on_bits__.tex b/section_Multivariate_polynomials_on_bits__.tex  index a0e96c0..ed457e6 100644  --- a/section_Multivariate_polynomials_on_bits__.tex  +++ b/section_Multivariate_polynomials_on_bits__.tex 
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\\  The $x$-degree of a term $x^iy^h$ in the two variables   $x,y$ is the value $i$, whereas $h$ is its $y$-degree.  In formulas To be more precise  $$  \deg_x(x^{i}y^h)=i,\;\deg_y(x^{i}y^h)=h  $$ 
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\\  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 $$x^2y^3z\, (i=2,h=3, l=1),\quad x^4\, (i=4,h=l=0),\quady^7 \, (i=l=0,h=7),\quad  z^9\, (i=h=0,l=9), \quad 1\, (i=h=l=0).$$ %  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, the following are polynomials $$x^2y^3+x+x^{10}+z, xy+z^{11}y^2, xyz+z^2+1 \textrm{  and }  x^3y^5z$$ (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  $$ 
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\\  The $x$-degree of a term $x^iy^hz^l$ in the two variables   $x,y$ is the value $i$, whereas $h$ is its $y$-degree and $l$ its $z$-degree.  In formulas To be more precise  $$  \deg_x(x^{i}y^hz^l)=i,\;\deg_y(x^{i}y^hz^l)=h,\; \deg_z(x^{i}y^hz^l)=l  $$ 
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\deg(x^iy^hz^l) \,=\, i+h+l\,.  $$  If we consider $xy^5z^3 \in \Fb[x,y,z]$,   we have then  $\deg_x(xy^5z^3)=1$, $\deg_y(xy^5z^3)=5$, $\deg_z(xy^5z^3)=3$, and $\deg(xy^5z^3)=9$. \\  The \emph{degree of a polynomial} $f\in \Fb[x,y,z] $ is the maximal degree of the monomials appearing  in $f$ with nonzero coefficient, so, if $f=x^3y+xy^6-z^2y\in \Fb[x,y,z]$, $\deg(f)=7$ and if $g=xz^3+0z^{12}$ then $\deg(g)=4$.