deletions | additions       diff --git a/section_Multivariate_polynomials_on_bits__.tex b/section_Multivariate_polynomials_on_bits__.tex  index b2fec30..032d816 100644  --- a/section_Multivariate_polynomials_on_bits__.tex  +++ b/section_Multivariate_polynomials_on_bits__.tex 
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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.   $x^{i} y^j$, for some $i,j$ in $\NN$.\\  For example, we can consider the following monomials $x^2y^3$ ($i=2,j=3$), $x^4$ ($i=4,j=0$), $y^7$ ($i=0,j=7$), $1$ ($i=j=0$). $$x^2y^3 (i=2,j=3), x^4 (i=4,j=0), y^7 (i=0,j=7), 1 (i=j=0).$$  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, $x^2y^3+x+x^{10}+y$, $xy+x^{11}y^2$ the following are polynomials  $$x^2y^3+x+x^{10}+y, xy+x^{11}y^2 \textrm{  and $x^3y^5$ } x^3y^5$$  (a monomial is also a polynomial). \\ Formally, a polynomial in the $2$ variables $x$ and $y$ and coefficients in $\Fb$ is any   expression of the form  $$