Any single variable polynomial in $GF(2)$ is reduced to first degree $x^n=x$. So all polynomials of $n$ variables will be like $x+y+xy$ We can transform any 3-SAT as a system of polynomial equations in $GF(2)$.
$x$ or $y$ is equivalent to $x+y+xy$ in $GF(2)$.
$x$ and $y$ is equivalent to $xy$ in $GF(2)$ and by adding some rules of solving for a variable in a polynomial equation in $GF(2)$ ($f(x)$ is denoted as the polynomial without $x$). We will have $xf(x)=1$ implies $x=1$; $f(x)=1$ $x(1+f(x))=f(x) \implies x=0$; $f(x) =0$.
And for the $x=f(x)$ for example $(x=y+z)$. Now after solving or writing $x$ in function of other variables we substitute in our system (Any other form of solving some variable will give us undetermined for example $xy=0$, $x$ is undetermined cause it can be true for any value)
We just add one more specification is that we won't just solve for a variable $x,y...$ but also for the product $xy, xyz...$ and after verifying that we can't solve for single variables so in a system of polynomial equations in $GF(2)$ we will create another system (system new) than we take each equation from the old system, substitute the variables that are inside from the equations that are from system new Than solve it Than do a back substitution in the new system And continue doing until there are no more equations or we ran into $1=0$. After that, we will have a simplified system each equation is written in terms of the next one So our system is simplified and can no more be and to find a solution we just substitute the last one My question is for 3-SAT after transforming it into a system of polynomial equations in $GF(2)$ and that the maximum of variable $M$s in each equation is 6 Can we, using this method solving in polynomial times? Thanks from advance