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

ForAll (∀)  ▪  Exists (∃)

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

Equation Structure

CoefficientList  ▪  CoefficientArrays  ▪  LogicalExpand