Solution set - Definition and Overview

In mathematics, a solution set for a collection of polynomials <math>\{f_i\}<math> over some ring <math>R<math> is defined to be the set <math>\{x\in R:\forall i\in I, f_i(x)=0\}<math>.

Examples

1. The solution set of <math>f(x):=x<math> over the real numbers is the set {0}.

2. For any non-zero polynomial <math>f<math> over the complex numbers in one variable, the solution set is made up of finitely many points. However, for a complex polynomial in more than one variable the solution set has no isolated points.

Remarks

In Algebraic Geometry solution sets are used to define the Zariski topology. See affine varieties.