# 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.

## 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 .