I came across linear equation $G(x,y) = g_k(x,y) l_k(x,y)$ mod $(y^{2^{k}})$ while reading factoring algorithm see section 3 for bivariate polynomials. I need to find the $G(x,y)$ and $l_k(x,y)$. Given : $g_k(x,y)$, where $deg_y(g_k(x,y)) < y^{2^{k}}$ and $deg_x(g_k(x,y)) < d $.Find : $G(x,y)$ and $l_k(x,y)$. (non trivial)I know that $G(x,y) = g_k(x,y) l_k(x,y)$ forms a linear system with $y^{2^{k +1}} \times 2d $ many equations and I can solve those equations. But I don't know how to solve correctly $G(x,y) = g_k(x,y) l_k(x,y)$ mod $(y^{2^{k}})$. Here is a problem, suppose after multiplying $g_k(x,y)$ and $l_k(x,y)$, I got a term $y^{2^{k}}$, but after doing mod operation this term is going to become $0$. So my question is, how to resolve this issue.
How to solve linear system with modulus operation?
matrices;linear algebra