CoincidentRootLoci -- A package for computations with coincident root loci

Description

This package accompanies the paper On the algebraic boundaries among typical ranks for real binary forms (see also Algebraic boundaries among typical ranks for real binary forms of arbitrary degree). It provides some tools that can be useful for working with symmetric tensors of dimension 2. Such tensors are bijectively associated with homogeneous polynomials in two variables, which are also called binary forms. One of the main methods is realrank, which uses QEPCAD to compute the real rank of binary forms defined over $\mathbb{Q}$.

Contributors

Maria Chiara Brambilla <[email protected]>

Author

Giovanni Staglianò <[email protected]>

Version

This documentation describes version 0.1.3 of CoincidentRootLoci.

Citation

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

@misc{CoincidentRootLociSource,
  title = {{CoincidentRootLoci: A \emph{Macaulay2} package. Version~0.1.3}},
  author = {Giovanni Staglian{\`o}},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- CoincidentRootLocus -- the class of all coincident root loci
Functions and commands
- apolar -- the apolar map
- coincidentRootLocus -- makes a coincident root locus
- projectiveJoin -- see CoincidentRootLocus * CoincidentRootLocus -- projective join of coincident root loci
- complexrank -- compute the complex rank
- CRL (missing documentation)
- generic -- see generic(CoincidentRootLocus) -- get the generic element
- polarDegrees -- polar degrees of a coincident root locus
- projectiveTangentSpace -- projective tangent space
- randomBinaryForm -- random homogeneous polynomial in two variables
- randomInCoisotropic -- see randomInCoisotropic(CoincidentRootLocus,ZZ) -- get a random element
- realrank -- compute the real rank
- realRankBoundary -- algebraic boundaries among typical ranks for real binary forms
- realroots -- real roots of a binary form
- recover -- recover the binary form from its apolar ideal
- supsets -- see supsets(CoincidentRootLocus) -- produce all the suploci
Methods
- apolar(ZZ,ZZ) -- see apolar -- the apolar map
- apolar(ZZ,ZZ,Ring) -- see apolar -- the apolar map
- apolar(RingElement) -- the apolar ideal
- apolar(RingElement,ZZ) -- homogeneous components of the apolar ideal
- chowForm(CoincidentRootLocus) -- Chow form of a coincident root locus
- codim(CoincidentRootLocus) -- compute the codimension
- coefficientRing(CoincidentRootLocus) -- get the coefficient ring
- coincidentRootLocus(List) -- see coincidentRootLocus -- makes a coincident root locus
- coincidentRootLocus(VisibleList,Ring) -- see coincidentRootLocus -- makes a coincident root locus
- CoincidentRootLocus * CoincidentRootLocus -- projective join of coincident root loci
- complexrank(RingElement) -- see complexrank -- compute the complex rank
- degree(CoincidentRootLocus) -- compute the degree
- dim(CoincidentRootLocus) -- compute the dimension
- dual(CoincidentRootLocus) -- the projectively dual to a coincident root locus
- generic(CoincidentRootLocus) -- get the generic element
- ideal(CoincidentRootLocus) -- the defining ideal of a coincident root locus
- isInCoisotropic(Ideal,CoincidentRootLocus) -- test membership in a coisotropic hypersurface
- isMember(RingElement,CoincidentRootLocus) -- test membership in a coincident root locus
- isSubset(CoincidentRootLocus,CoincidentRootLocus) -- whether one object is a subset of another
- map(CoincidentRootLocus) -- the map associated to a coincident root locus
- partition(CoincidentRootLocus) -- the partition associated to a coincident root locus
- polarDegrees(CoincidentRootLocus) -- see polarDegrees -- polar degrees of a coincident root locus
- projectiveTangentSpace(CoincidentRootLocus,RingElement) -- see projectiveTangentSpace -- projective tangent space
- random(CoincidentRootLocus) -- get a random element
- randomBinaryForm(ZZ) -- see randomBinaryForm -- random homogeneous polynomial in two variables
- randomBinaryForm(ZZ,Ring) -- see randomBinaryForm -- random homogeneous polynomial in two variables
- randomBinaryForm(ZZ,Thing,Thing) -- see randomBinaryForm -- random homogeneous polynomial in two variables
- randomBinaryForm(ZZ,Thing,Thing,Ring) -- see randomBinaryForm -- random homogeneous polynomial in two variables
- randomInCoisotropic(CoincidentRootLocus,ZZ) -- get a random element
- realrank(RingElement) -- see realrank -- compute the real rank
- realRankBoundary(ZZ,ZZ) -- see realRankBoundary -- algebraic boundaries among typical ranks for real binary forms
- realRankBoundary(ZZ,ZZ,Ring) -- see realRankBoundary -- algebraic boundaries among typical ranks for real binary forms
- realroots(RingElement) -- see realroots -- real roots of a binary form
- recover(Ideal) -- see recover -- recover the binary form from its apolar ideal
- recover(RingElement,RingElement) -- see recover -- recover the binary form from its apolar ideal
- ring(CoincidentRootLocus) -- get the ring of a coincident root locus
- singularLocus(CoincidentRootLocus) -- the singular locus of a coincident root locus
- subsets(CoincidentRootLocus) -- produce all the subloci
- supsets(CoincidentRootLocus) -- produce all the suploci
- switch(Ideal) -- see switch(RingElement)
- switch(List) -- see switch(RingElement)
- switch(RingElement)
Symbols
- QepcadOptions -- set the number of cells in the garbage collected space

For the programmer

The object CoincidentRootLoci is a package, defined in CoincidentRootLoci.m2, with auxiliary files in CoincidentRootLoci/.

The source of this document is in CoincidentRootLoci/documentation.m2:7:0.