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Reduce

Reduce[expr,vars]

reduces the statement expr by solving equations or inequalities for vars and eliminating quantifiers.

Reduce[expr,vars,dom]

does the reduction over the domain dom. Common choices of dom are Reals, Integers, and Complexes.

Details and Options

The statement expr can be any logical combination of:

	lhs==rhs	equations
	lhs!=rhs	inequations
	lhs>rhs or lhs>=rhs	inequalities
	expr∈dom	domain specifications
	{x,y,…}∈reg	region specification
	ForAll[x,cond,expr]	universal quantifiers
	Exists[x,cond,expr]	existential quantifiers

The result of Reduce[expr,vars] always describes exactly the same mathematical set as expr.
Reduce[{expr₁,expr₂,…},vars] is equivalent to Reduce[expr₁&&expr₂&&…,vars].
Reduce[expr,vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
Reduce[expr,vars,dom] restricts all variables and parameters to belong to the domain dom.
If dom is Reals, or a subset such as Integers or Rationals, then all constants and function values are also restricted to be real.
Reduce[expr&&vars∈Reals,vars,Complexes] performs reductions with variables assumed real, but function values allowed to be complex.
Reduce[expr,vars,Integers] reduces Diophantine equations over the integers.
Reduce[…,x∈reg,Reals] constrains x to be in the region reg. The different coordinates for x can be referred to using Indexed[x,i].
Reduce[expr,{x₁,x₂,…},…] effectively writes expr as a combination of conditions on x₁, x₂, … where each condition involves only the earlier .
Algebraic variables in expr free of the and of each other are treated as independent parameters.
Applying LogicalExpand to the results of Reduce[expr,…] yields an expression of the form , where each of the can be thought of as representing a separate component in the set defined by expr.
The may not be disjoint and may have different dimensions. After LogicalExpand, each of the has the form .
Without LogicalExpand, Reduce by default returns a nested collection of conditions on the , combined alternately by Or and And on successive levels.
When expr involves only polynomial equations and inequalities over real or complex domains, then Reduce can always in principle solve directly for all the .
When expr involves transcendental conditions or integer domains, Reduce will often introduce additional parameters in its results.
When expr involves only polynomial conditions, Reduce[expr,vars,Reals] gives a cylindrical algebraic decomposition of expr.
Reduce can give explicit representations for solutions to all linear equations and inequalities over the integers and can solve a large fraction of Diophantine equations described in the literature.
When expr involves only polynomial conditions over real or complex domains, Reduce[expr,vars] will always eliminate quantifiers, so that quantified variables do not appear in the result.
The following options can be given:

Backsubstitution	False	whether to give results unwound by backsubstitution »
Cubics	False	whether to use explicit radicals to solve all cubics »
GeneratedParameters	C	how to name parameters that are generated »
Modulus	0	modulus to assume for integers »
Quartics	False	whether to use explicit radicals to solve all quartics »

Reduce[expr,{x₁,x₂,…},Backsubstitution->True] yields a form in which values from equations generated for earlier are backsubstituted so that the conditions for a particular have only minimal dependence on earlier . »

Examples

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Basic Examples (4)

Reduce equations and inequalities:

Use specific domains:

Reduce a quantified expression:

Reduce with geometric region constraints:

Scope (83)

Basic Uses (5)

Find an explicit description of the solution set of a system of equations:

Use ToRules and ReplaceRepeated (//.) to list the solutions:

Find an explicit description of the solution set of a system of inequalities:

Find solutions over specified domains:

The solution set may depend on symbolic parameters:

Representing solutions may require introduction of new parameters:

List the first 10 solutions:

Complex Domain (16)

A linear system:

A univariate polynomial equation:

A multivariate polynomial equation:

Systems of polynomial equations and inequations can always be reduced:

A quantified polynomial system:

An algebraic system:

Transcendental equations solvable in terms of inverse functions:

In this case there is no solution:

Equations involving elliptic functions:

Equations solvable using special function zeros:

Solving this system does not require the Riemann hypothesis:

Elementary function equation in a bounded region:

Holomorphic function equation in a bounded region:

Here Reduce finds some solutions but is not able to prove there are no other solutions:

Equation with a purely imaginary period over a vertical stripe in the complex plane:

Doubly periodic transcendental equation:

A system of transcendental equations solvable using inverse functions:

A square system of analytic equations over a bounded box:

Real Domain (26)

A linear system:

A univariate polynomial equation:

A univariate polynomial inequality:

A multivariate polynomial equation:

A multivariate polynomial inequality:

Systems of polynomial equations and inequalities can always be reduced:

A quantified polynomial system:

An algebraic system:

Piecewise equations:

Piecewise inequalities:

Transcendental equations, solvable using inverse functions:

Transcendental inequalities, solvable using inverse functions:

Inequalities involving elliptic functions:

Transcendental equation, solvable using special function zeros:

Transcendental inequality, solvable using special function zeros:

Exp-log equations:

High-degree sparse polynomial equation:

Algebraic equation involving high-degree radicals:

Equation involving irrational real powers:

Exp-log inequality:

Elementary function equation in a bounded interval:

Holomorphic function equation in a bounded interval:

Meromorphic function inequality in a bounded interval:

Periodic elementary function equation over the reals:

Transcendental systems solvable using inverse functions:

Systems exp-log in the first variable and polynomial in the other variables:

Quantified system:

Systems elementary and bounded in the first variable and polynomial in the other variables:

Quantified system:

Systems analytic and bounded in the first variable and polynomial in the other variables:

Quantified system:

Square systems of analytic equations over bounded regions:

Integer Domain (13)

Linear system of equations:

A linear system of equations and inequalities:

A univariate polynomial equation:

A univariate polynomial inequality:

Binary quadratic equations:

A Thue equation:

A sum of squares equation:

The Pythagorean equation:

A bounded system of equations and inequalities:

A high-degree system with no solution:

Transcendental Diophantine systems:

A polynomial system of congruences:

Diophantine equations with irrational coefficients:

Modular Domains (5)

A linear system:

A univariate polynomial equation:

A multivariate polynomial equation:

A system of polynomial equations and inequations:

Reduce a quantified polynomial system:

Finite Field Domains (4)

Univariate equations:

Systems of linear equations:

Systems of polynomial equations:

Systems involving quantifiers:

Mixed Domains (4)

Mixed real and complex variables:

Find real values of and complex values of for which is real and less than :

Reduce an inequality involving Abs[x]:

Plot the solution set:

Mixed integer and real variables:

Geometric Regions (10)

Constrain variables to basic geometric regions in 2D:

Plot the solution:

Constrain variables to basic geometric regions in 3D:

Plot the solution:

Project a 3D region onto the - plane:

Plot the projection:

An implicitly defined region:

A parametrically defined region:

Derived regions:

The solution of restricted to the intersection:

Eliminate quantifiers over a Cartesian product of regions:

Regions dependent on parameters:

A condition for :

Use to specify that is a vector in :

In this case is a vector in :

Options (6)

Backsubstitution (1)

Since y appears after x in the variable list, Reduce may use x to express the solution for y:

With Backsubstitution->True, Reduce gives explicit numeric values for y:

Cubics (1)

By default, Reduce does not use general formulas for solving cubics in radicals:

With Cubics->True, Reduce solves all cubics in terms of radicals:

GeneratedParameters (1)

Reduce may introduce new parameters to represent the solution:

Use GeneratedParameters to control how the parameters are generated:

Modulus (1)

Solve equations over the integers modulo 9:

Quartics (1)

By default, Reduce does not use general formulas for solving quartics in radicals:

With Quartics->True, Reduce solves all quartics in terms of radicals:

WorkingPrecision (1)

Finding the solution with exact computations takes a long time:

With WorkingPrecision->100, Reduce finds a solution fast, but it may be incorrect:

Applications (9)

Basic Applications (1)

Prove geometric inequalities for , , and sides of a triangle:

Prove an inequality for triangles:

Prove an inequality for acute triangles:

Polynomial Root Problems (1)

Find conditions for a quartic to have all roots equal:

Using quantifier elimination:

Using Subresultants:

Parametrization Problems (1)

Plot a space curve given by an implicit description:

Plot the projection of the space curve on the - plane:

Integer Problems (3)

Find a Pythagorean triple:

Find a sequence of Pythagorean triples:

Find how to pay $2.27 postage with 10-, 23- and 37-cent stamps:

The same task can be accomplished with IntegerPartitions:

Show that there are only five regular polyhedrons:

Each face for a regular -gon contributes edges, but they are shared, so they are counted twice:

Each face for a regular -gon contributes vertices, but they are shared, so they are counted times:

Using Euler's formula , find the number of faces:

For this last formula to be well defined, the denominator needs to be positive and an integer:

Hence the following five cases:

Compare this to the actual counts in PolyhedronData:

Geometry Problems (3)

The region ℛ is a subset of  if is true. Show that Disk[{0,0},{2,1}] is a subset of Rectangle[{-2,-1},{2,1}]:

Plot it:

Show that Cylinder[]⊆Ball[{0,0,0},2]:

Plot it:

For a finite point set , the Voronoi cell for a point can be defined by , which corresponds to all points closer to than any other point for . Find a simple formula for a Voronoi cell, using Reduce: