Implementing arithmetic over algebraic numbers A tutorial for Lazard's lifting scheme in CAD

Implementing techniques from computer algebra often requires a multitude of foundational algorithms that are neither easy to understand nor to implement. Despite great interest from other communities, the difficulty to implement novel techniques from computer algebra proves to be a significant hindrance, especially, when a modern computer algebra system cannot be used. We tackle cylindrical algebraic decomposition (CAD) as one such example. CAD can be, for example, applied in satisfiability modulo theories solving for nonlinear real arithmetic. However, a recent advance in CAD, the Lazard's lifting scheme, requires additional algebraic techniques that are neither available in these solvers, nor as stand-alone libraries. We close this gap by showing how to use the CoCoALib library to implement Lazard's lifting scheme outside of a modern computer algebra system like Maple.