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

Description

This package provides general tools for elimination and solving systems of polynomial equations, with a particular emphasis on real solutions.

Acknowledgement

This package is an updated version the realroots package by Grayson and Sottile.

Authors

Certification

Version 1.0 of this package was accepted for publication in volume 14 of Journal of Software for Algebra and Geometry on 2024-03-18, in the article Real solutions to systems of polynomial equations in Macaulay2 (DOI: 10.2140/jsag.2024.14.87). That version can be obtained from the journal.

Version

This documentation describes version 1.1 of RealRoots.

Citation

If you have used this package in your research, please cite it as follows:

@misc{RealRootsSource,
  title = {{RealRoots: A \emph{Macaulay2} package. Version~1.1}},
  author = {Jordy Lopez Garcia and Kelly Maluccio and Frank Sottile and Thomas Yahl},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

@article{RealRootsArticle,
  title = {{Real solutions to systems of polynomial equations in \emph{Macaulay2}}},
  author = {Jordy Lopez Garcia and Kelly Maluccio and Frank Sottile and Thomas Yahl},
  journal = {Journal of Software for Algebra and Geometry},
  volume = {14},
  year = {2024},
}

Exports

Functions and commands
- budanFourierBound -- a bound for the number of real roots with multiplicity 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 -- the principal submatrix of the Hurwitz matrix of a univariate polynomial
- isHurwitzStable -- determines if a univariate polynomial with rational coefficients is Hurwitz-stable
- minimalPolynomial -- the minimal polynomial of an element of an Artinian ring
- rationalUnivariateRepresentation -- the rational univariate representation of a zero-dimensional ideal in a polynomial ring
- realCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
- realRootIsolation -- a list that isolates the real roots of a univariate polynomial with rational coefficients
- regularRepresentation -- the regular representation of an element of an Artinian ring
- signature -- the signature of a symmetric matrix with rational entries
- sturmCount -- the number of real roots of a univariate polynomial with rational coefficients
- sturmSequence -- the Sturm sequence of a univariate polynomial with rational coefficients
- sylvesterCount -- a signed count of the real roots of a univariate polynomial with rational coefficients
- sylvesterSequence -- the Sylvester sequence of two univariate polynomials with rational coefficients
- traceCount -- the degree of the reduced scheme of an Artinian ring
- traceForm -- the trace symmetric form of a 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 with multiplicity of a univariate polynomial with rational coefficients
- budanFourierBound(RingElement,QQ,QQ) -- see budanFourierBound -- a bound for the number of real roots with multiplicity 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 -- the principal submatrix of the Hurwitz matrix of a univariate polynomial
- hurwitzMatrix(RingElement,ZZ) -- see hurwitzMatrix -- the principal submatrix of the Hurwitz matrix of a univariate polynomial
- isHurwitzStable(RingElement) -- see isHurwitzStable -- determines if a univariate polynomial with rational coefficients 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 in a polynomial ring
- rationalUnivariateRepresentation(QuotientRing) -- see rationalUnivariateRepresentation -- the rational univariate representation of a zero-dimensional ideal in a polynomial ring
- realCount(Ideal) -- see realCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
- realCount(List) -- see realCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
- realCount(QuotientRing) -- see realCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
- realCount(RingElement) -- see realCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)
- realRootIsolation(RingElement,QQ) -- see realRootIsolation -- a list that isolates the real roots of a univariate polynomial with rational coefficients
- regularRepresentation(RingElement) -- see regularRepresentation -- the regular representation of an element of an Artinian ring
- regularRepresentation(RingElement,Ideal) -- see regularRepresentation -- the regular representation of an element of an Artinian ring
- signature(Matrix) -- see signature -- the signature of a symmetric matrix with rational entries
- sturmCount(RingElement) -- see sturmCount -- the number of real roots of a univariate polynomial with rational coefficients
- sturmCount(RingElement,QQ,QQ) -- see sturmCount -- the number of real roots of a univariate polynomial with rational coefficients
- sturmSequence(RingElement) -- see sturmSequence -- the Sturm sequence of a univariate polynomial with rational coefficients
- sylvesterCount(RingElement,RingElement) -- see sylvesterCount -- a signed count of the real roots of a univariate polynomial with rational coefficients
- sylvesterCount(RingElement,RingElement,QQ,QQ) -- see sylvesterCount -- a signed count of the real roots of a univariate polynomial with rational coefficients
- sylvesterSequence(RingElement,RingElement) -- see sylvesterSequence -- the Sylvester sequence of two univariate polynomials with rational coefficients
- traceCount(Ideal) -- see traceCount -- the degree of the reduced scheme of an Artinian ring
- traceCount(List) -- see traceCount -- the degree of the reduced scheme of an Artinian ring
- traceCount(QuotientRing) -- see traceCount -- the degree of the reduced scheme of an Artinian ring
- traceCount(RingElement) -- see traceCount -- the degree of the reduced scheme of an Artinian ring
- traceForm(RingElement) -- see traceForm -- the trace symmetric form of a polynomial in an Artinian ring
- traceForm(RingElement,Ideal) -- see traceForm -- the trace symmetric form of a polynomial in 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
- Multiplicity

For the programmer

The object RealRoots is a package, defined in RealRoots.m2.

The source of this document is in RealRoots.m2:634:0.