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 zerodimensional 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 zerodimensional 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