# Lazily solving equations in powerseries rings over finite fields

Hello,

I am currently trying to compute coefficients in implicitly defined powerserieses over finite fields. The first is a linear equation of the form a*x+b=0, with a, b given as multivariate lazy powerserieses over the field with 4 elements, and from theoretical considerations I know that there is a solution x=1+... . Even though just writing x=b/a yields an exception, I at least have a more or less efficient way of computing x "by hand", but i would like to be able to define it as a lazy series.

The second application is a bit harder: I want to solve for a powerseries l in one variable over the field with 4 elements that fullfills a system of nonlinear equations, and from theoretical considerations I know that there is a unique solution. My current method is defining l as a powerseries over the infinite polynomial ring over the filed with 4 elements, with coefficients the infinitely many polynomial generators, plugging that into the equations, and solving degree be degree, which takes extremly long...

Does anyone have a suggestion on how to implement this kind of solving as a lazy series?

The question is too abstract. Please provide an actual example.