deletions | additions       diff --git a/section_Multivariate_polynomials_on_bits__.tex b/section_Multivariate_polynomials_on_bits__.tex  index 6f2cfcb..02a96fd 100644  --- a/section_Multivariate_polynomials_on_bits__.tex  +++ b/section_Multivariate_polynomials_on_bits__.tex 
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As for section \ref{Sec:Polynomials}, we start by defining terms (or monomials).  \\  A \emph{term} or \emph{monomial} in $n$ variables $x_1,...x_n$ is a product of powers of $x_1,...x_n$, i.e.   $x_1^{a_1}\cdots x_n^{a_n}$ $x_1^{i_1}\cdots x_n^{i_n}$  for some $a_1,...,a_n $i_1,...,i_n  \in \NN$.\\ 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.  \\  With terms, we can define polynomials in $n$ variables $x_1,...x_n$ and coefficients in $\Fb$ (i.e. multivariate polynomials in the field of bits) as expressions of the form 
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When (as we will usually do) we will deal with two or three variables, we will denote them as $x,y$ or $x,y,z$, so for example $xy+1\in \Fb[x,y]$, $x^4y-z^3 \in \Fb[x,y,z]$.  \\  For $1 \leq j\leq n$, the \emph{$j$-degree} of a term in $n$ variables $x_1^{i_1}\cdots x_n^{i_n}$ is the value $i_j$. In formulas   $$deg_j(x_1^{i_1}\cdots x_n^{i_n}).$$ x_n^{i_n})=i_j.$$  The \emph{total degree} (or, simply, degree)   of a term in $n$ variables $x_1^{i_1}\cdots x_n^{i_n}$ is the sum of all $deg_j$ for all $1 \leq j\leq n$, i.e.  $$deg(x_1^{i_1}\cdots x_n^{i_n}) =\sum_{j=1}^n deg_j(x_1^{i_1}\cdots x_n^{i_n}) =i_1+...+i_n.$$