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RealRoots -- Package for exploring, counting and locating real solutions to polynomial systems

Description

The purpose of this package is to provide general tools for elimination and solving systems of polynomial equations.

Authors

Version

This documentation describes version 0.1 of RealRoots.

Source code

The source code from which this documentation is derived is in the file RealRoots.m2.

Exports

  • Functions and commands
    • BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • characteristicPolynomial -- the characteristic polynomial of a matrix or the characteristic polynomial of the regular representation of a polynomial
    • HurwitzMatrix -- a specified principle submatrix of the Hurwitz matrix of a univariate polynomial
    • isHurwitzStable -- determines whether or not a rational univariate polynomial is Hurwitz stable
    • minimalPolynomial -- the minimal polynomial of an element of an Artinian ring
    • rationalUnivariateRepresentation -- the rational univariate representation of a zero-dimensional ideal
    • realRootIsolation -- a list that isolates the real roots of a rational univariate polynomial
    • regularRepresentation -- the regular representation of a rational polynomial
    • SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmSequence -- the Sturm sequence of a rational univariate polynomial
    • SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterSequence -- the Sylvester sequence of two rational univariate polynomials
    • traceCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
    • traceForm -- the trace quadratic form of a rational polynomial in an Artinian ring
    • univariateEliminant -- the univariate eliminant of an element of an Artinian ring
    • variations -- the number of sign changes of an ordered list of numbers
  • Methods
    • BudanFourierBound(RingElement) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,InfiniteNumber,InfiniteNumber) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,InfiniteNumber,QQ) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,InfiniteNumber,RR) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,InfiniteNumber,ZZ) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,QQ,InfiniteNumber) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,QQ,QQ) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,QQ,RR) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,QQ,ZZ) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,RR,InfiniteNumber) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,RR,QQ) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,RR,RR) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,RR,ZZ) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,ZZ,InfiniteNumber) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,ZZ,QQ) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,ZZ,RR) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • BudanFourierBound(RingElement,ZZ,ZZ) -- see BudanFourierBound -- a bound for the number of real roots of a univariate polynomial with rational coefficients
    • characteristicPolynomial(Matrix) -- see characteristicPolynomial -- the characteristic polynomial of a matrix or the characteristic polynomial of the regular representation of a polynomial
    • characteristicPolynomial(RingElement) -- see characteristicPolynomial -- the characteristic polynomial of a matrix or the characteristic polynomial of the regular representation of a polynomial
    • characteristicPolynomial(RingElement,Ideal) -- see characteristicPolynomial -- the characteristic polynomial of a matrix or the characteristic polynomial of the regular representation of a polynomial
    • HurwitzMatrix(RingElement) -- see HurwitzMatrix -- a specified principle submatrix of the Hurwitz matrix of a univariate polynomial
    • HurwitzMatrix(RingElement,ZZ) -- see HurwitzMatrix -- a specified principle submatrix of the Hurwitz matrix of a univariate polynomial
    • isHurwitzStable(RingElement) -- see isHurwitzStable -- determines whether or not a rational univariate polynomial is Hurwitz stable
    • minimalPolynomial(RingElement) -- see minimalPolynomial -- the minimal polynomial of an element of an Artinian ring
    • minimalPolynomial(RingElement,Ideal) -- see minimalPolynomial -- the minimal polynomial of an element of an Artinian ring
    • rationalUnivariateRepresentation(Ideal) -- see rationalUnivariateRepresentation -- the rational univariate representation of a zero-dimensional ideal
    • realRootIsolation(RingElement,QQ) -- see realRootIsolation -- a list that isolates the real roots of a rational univariate polynomial
    • realRootIsolation(RingElement,RR) -- see realRootIsolation -- a list that isolates the real roots of a rational univariate polynomial
    • realRootIsolation(RingElement,ZZ) -- see realRootIsolation -- a list that isolates the real roots of a rational univariate polynomial
    • regularRepresentation(RingElement) -- see regularRepresentation -- the regular representation of a rational polynomial
    • regularRepresentation(RingElement,Ideal) -- see regularRepresentation -- the regular representation of a rational polynomial
    • SturmCount(RingElement) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,InfiniteNumber,InfiniteNumber) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,InfiniteNumber,QQ) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,InfiniteNumber,RR) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,InfiniteNumber,ZZ) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,QQ,InfiniteNumber) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,QQ,QQ) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,QQ,RR) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,QQ,ZZ) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,RR,InfiniteNumber) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,RR,QQ) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,RR,RR) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,RR,ZZ) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,ZZ,InfiniteNumber) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,ZZ,QQ) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,ZZ,RR) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmCount(RingElement,ZZ,ZZ) -- see SturmCount -- the number of real roots of a rational univariate polynomial
    • SturmSequence(RingElement) -- see SturmSequence -- the Sturm sequence of a rational univariate polynomial
    • SylvesterCount(RingElement,RingElement) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,InfiniteNumber,InfiniteNumber) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,InfiniteNumber,QQ) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,InfiniteNumber,RR) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,InfiniteNumber,ZZ) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,QQ,InfiniteNumber) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,QQ,QQ) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,QQ,RR) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,QQ,ZZ) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,RR,InfiniteNumber) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,RR,QQ) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,RR,RR) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,RR,ZZ) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,ZZ,InfiniteNumber) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,ZZ,QQ) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,ZZ,RR) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterCount(RingElement,RingElement,ZZ,ZZ) -- see SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials
    • SylvesterSequence(RingElement,RingElement) -- see SylvesterSequence -- the Sylvester sequence of two rational univariate polynomials
    • traceCount(Ideal) -- see traceCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
    • traceCount(List) -- see traceCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
    • traceCount(QuotientRing) -- see traceCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
    • traceCount(RingElement) -- see traceCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
    • traceForm(RingElement) -- see traceForm -- the trace quadratic form of a rational polynomial in an Artinian ring
    • traceForm(RingElement,Ideal) -- see traceForm -- the trace quadratic form of a rational polynomial in an Artinian ring
    • univariateEliminant(RingElement) -- see univariateEliminant -- the univariate eliminant of an element of an Artinian ring
    • univariateEliminant(RingElement,Ideal) -- see univariateEliminant -- the univariate eliminant of an element of an Artinian ring
    • variations(List) -- see variations -- the number of sign changes of an ordered list of numbers
  • Symbols

For the programmer

The object RealRoots is a package.