Solving the ARE symbolically

Methods from computer algebra, mostly so called Grobner bases (gb) from commutative algebra, are used to solve the algebraic Riccati equation (ARE) symbolically. The methods suggested allow us to track the influence of parameters in the system or penalty matrices on the solution. Some nontrivial aspects arise when addressing the problem from the point of view commutative algebra, for example the original equations are rational, not polynomial. We explain how this can be dealt with rather easily. Some methods for lowering the computational complexity are suggested and different methods are compared regarding efficiency. Preprocessing of the equations before applying gb can make computations more efficient