Polynomial Systems
The Wolfram Language's handling of polynomial systems is a tour de force of algebraic computation. Building on mathematical results spanning more than a century, the Wolfram Language for the first time implements complete efficient reduction of polynomial equation and inequality systems—making possible industrial-strength generalized algebraic geometry for many new applications.
Solving & Reducing
Solve — find generic solutions for variables
Reduce — reduce systems of equations and inequalities to canonical form
Complexes, Reals, Integers — domains for variables
Eliminating Variables
Eliminate — eliminate variables between equations
SolveAlways — solve for parameter values that make equations always hold
GroebnerBasis ▪ Resultant ▪ Discriminant ▪ Subresultants
Quantifier Elimination
Resolve — eliminate general quantifiers
Reduce — eliminate quantifiers and reduce the results
Structure of Solution Sets
SemialgebraicComponentInstances ▪ CylindricalDecomposition ▪ GenericCylindricalDecomposition ▪ CylindricalDecompositionFunction ▪ FindInstance
Numerical Solutions
NSolve — solve systems of polynomial equations
Optimization »
Minimize ▪ Maximize ▪ NMinimize ▪ NMaximize
Visualization
ContourPlot — curve or curves defined by equation in x and y
ContourPlot3D — surface defined by equation in x, y and z
RegionPlot, RegionPlot3D — regions defined by inequalities