deletions | additions       diff --git a/section_Multivariate_polynomials_on_bits__.tex b/section_Multivariate_polynomials_on_bits__.tex  index 060dfb0..6194978 100644  --- a/section_Multivariate_polynomials_on_bits__.tex  +++ b/section_Multivariate_polynomials_on_bits__.tex 
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For example, we can consider the following monomials $x^2y^3$ ($i=2,j=3$), $x^4$ ($i=4,j=0$), $y^7$ ($i=0,j=7$), $1$ ($i=j=0$).  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, $x^2y^3+x+x^10+y$, $xy+x^11y^2$ $x^2y^3+x+x^{10}+y$, $xy+x^{11}y^2$  and $x^3y^5$ (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  $$ 
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\begin{Exercise}\label{operazionibivar}  Compute the following sums and products in $\Fb[x,y]$ and find the degree of the final result:  \begin{itemize}  \item $(xy^3+y)+(xy^3+x^2z+)$ $(xy^3+y)+(xy^3+x^2)$  \item $(x+y+y^3)(x+y+y^3)$  \item $\big((x+y)(xy^3+y)\big)+(xy+y^2+1)$  \item $\big((x+1)(y+y^3x)\big)+\big((xy+y^2+y+1)(x+1)\big)$