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

Related Tech Notes

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• Structural Operations on Polynomials
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• Algebraic Operations on Polynomials
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• Polynomials over Algebraic Number Fields
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• Polynomials Modulo Primes
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• Complex Polynomial Systems
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• Real Polynomial Systems
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• Diophantine Polynomial Systems

Related Guides

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• Polynomial Equations
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• Diophantine Equations
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• Linear Algebra & Matrices
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• Algebraic Number Fields

Related Links

• Demonstrations related to Polynomial Systems (Wolfram Demonstrations Project)