I am using the quotient ring $\mathbb{GF}_2[x]/(x^2 + x)$ (the boolean ring) and I have a system of polynomials in $n$ variables with arbitrary degree. I'd like to find a bijective transformation whereby the resulting system contains no cubic terms, but does contain linear and quadratic terms, and may contain terms of degree $>3$. Ideally, I'd like to use a similar method to transform the resulting system again to remove the degree 4 terms (while not introducing any degree 3 terms, and continuing to have linear & quadratic terms).
Example:
$$ \begin{align} f_1 &= x_1x_2x_3 + x_1x_2 + x_1x_4 + x_3 \\ f_2 &= x_1x_3x_4 + x_2x_4 + x_1 \\ \end{align} $$
Transform this system using
$$ \begin{align} x_1 &= y_1 \\ x_2 &= y_2 \\ x_3 &= y_3y_4 + y_1 + y_3 \\ x_4 &= y_2y_3 + y_4 \\ \end{align} $$
Giving you
$$ \begin{align} f_1' &= y_1y_2y_3y_4 + y_1y_4 + y_3y_4 + y_1 + y_3 \\ f_2' &= y_1y_2y_3y_4 + y_2y_3 + y_1y_4 + y_2y_4 + y_1 \\ \end{align} $$
Note that $f_1'$ and $f_2'$ contain no cubic terms. This example was created by guessing values until the desired result was achieved. Obviously I'd like some sort of algorithm to generate the transformation. The degree of the transformation is unimportant (I used degree 2 - which is a minimum here).