More generally, the solution set to an arbitrary collection E of
relations (Ei) (i varying in some index set I) for a collection of unknowns , supposed to take values in respective spaces , is the set S of all solutions to the relations E, where a solution is a family of values such that substituting by in the collection E makes all relations "true".
The above meaning is a special case of this one, if the set of polynomials fi if interpreted as the set of equations fi(x)=0.
Examples
The solution set for E = { x+y = 0 } with respect to is S = { (a,−a) : a ∈ R }.
The solution set for E = { x+y = 0 } with respect to is S = { −y }. (Here, y is not "declared" as an unknown, and thus to be seen as a
parameter on which the equation, and therefore the solution set, depends.)
The solution set for with respect to is the interval S = [0,2] (since is undefined for negative values of x).
The solution set for with respect to is S = 2πZ (see
Euler's identity).