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 systemsmaking possible industrial-strength generalized algebraic geometry for many new applications.

ReferenceReference

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  ▪  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