SparseResultants -- computations with sparse resultants
Description
This package provides the methods sparseResultant and sparseDiscriminant to calculate sparse resultants and sparse discriminants. As an application, the method determinant(MultidimensionalMatrix) is also provided. See also the package Resultants, which includes methods to calculate dense resultants and dense discriminants.
For the definitions, see one of the following books:
(1) Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky;
(2) Using Algebraic Geometry, by David A. Cox, John Little, Donal O'shea.
Caveat
Currently, most of the algorithms implemented are based on elimination via Gröbner basis methods. This should be significantly improved by implementing more specialized algorithms (see e.g. [2, Chapter 7,Section 6]).Author
- Giovanni Staglianò <giovannistagliano@gmail.com>
Certification 
Version 1.1 of this package was accepted for publication in volume 11 of The Journal of Software for Algebra and Geometry on 5 May 2021, in the article A package for computations with sparse resultants. That version can be obtained from the journal or from the Macaulay2 source code repository.
Version
This documentation describes version 1.2 of SparseResultants.
Source code
The source code from which this documentation is derived is in the file SparseResultants.m2.
Exports
-
Types
- MultidimensionalMatrix -- the class of all multidimensional matrices
- SparseDiscriminant -- the class of all sparse discriminants
- SparseResultant -- the class of all sparse resultants
-
Functions and commands
- degreeDeterminant -- degree of the hyperdeterminant of a generic multidimensional matrix
- denseDiscriminant -- dense discriminant (classical discriminant)
- denseResultant -- dense resultant (classical resultant)
- exponentsMatrix -- exponents in one or more polynomials
- flattening -- flattening of a multidimensional matrix
- genericLaurentPolynomials -- generic (Laurent) polynomials
- genericMultidimensionalMatrix -- make a generic multidimensional matrix of variables
- genericMultihomogeneousPolynomial -- generic multi-homogeneous polynomial
- genericSkewMultidimensionalMatrix -- make a generic skew symmetric multidimensional matrix of variables
- genericSymmetricMultidimensionalMatrix -- make a generic symmetric multidimensional matrix of variables
- multidimensionalMatrix -- make a multidimensional matrix
- permute -- permute the dimensions of a multidimensional matrix
- randomMultidimensionalMatrix -- random multidimensional matrix
- reverseShape -- reverse the dimensions of a multidimensional matrix
- shape -- shape of a multidimensional matrix
- sortShape -- sort the dimensions of a multidimensional matrix
- sparseDiscriminant -- sparse discriminant (A-discriminant)
- sparseResultant -- sparse resultant (A-resultant)
- sylvesterMatrix -- Sylvester-type matrix for the hyperdeterminant of a matrix of boundary shape
-
Methods
- char(SparseDiscriminant) -- get the characteristic
- char(SparseResultant) -- get the characteristic
- "degreeDeterminant(List)" -- see degreeDeterminant -- degree of the hyperdeterminant of a generic multidimensional matrix
- "denseDiscriminant(RingElement)" -- see denseDiscriminant -- dense discriminant (classical discriminant)
- "denseDiscriminant(ZZ,ZZ)" -- see denseDiscriminant -- dense discriminant (classical discriminant)
- "denseResultant(Thing)" -- see denseResultant -- dense resultant (classical resultant)
- determinant(MultidimensionalMatrix) -- hyperdeterminant of a multidimensional matrix
- dim(MultidimensionalMatrix) -- dimension of a multidimensional matrix
- entries(MultidimensionalMatrix) -- lists the entries of a multidimensional matrix
- exponents(SparseDiscriminant) -- get the matrix of exponents
- exponents(SparseResultant) -- get the matrices of exponents
- "exponentsMatrix(Sequence)" -- see exponentsMatrix -- exponents in one or more polynomials
- "flattening(List,MultidimensionalMatrix)" -- see flattening -- flattening of a multidimensional matrix
- "flattening(ZZ,MultidimensionalMatrix)" -- see flattening -- flattening of a multidimensional matrix
- "genericLaurentPolynomials(Sequence)" -- see genericLaurentPolynomials -- generic (Laurent) polynomials
- "genericMultidimensionalMatrix(VisibleList)" -- see genericMultidimensionalMatrix -- make a generic multidimensional matrix of variables
- "genericMultihomogeneousPolynomial(VisibleList,VisibleList)" -- see genericMultihomogeneousPolynomial -- generic multi-homogeneous polynomial
- "genericSkewMultidimensionalMatrix(ZZ,ZZ)" -- see genericSkewMultidimensionalMatrix -- make a generic skew symmetric multidimensional matrix of variables
- "genericSymmetricMultidimensionalMatrix(ZZ,ZZ)" -- see genericSymmetricMultidimensionalMatrix -- make a generic symmetric multidimensional matrix of variables
- "multidimensionalMatrix(List)" -- see multidimensionalMatrix -- make a multidimensional matrix
- MultidimensionalMatrix * MultidimensionalMatrix -- product of multidimensional matrices
- MultidimensionalMatrix + MultidimensionalMatrix -- sum of two multidimensional matrices of the same shape
- MultidimensionalMatrix - MultidimensionalMatrix -- difference of two multidimensional matrices of the same shape
- MultidimensionalMatrix == MultidimensionalMatrix -- equality of two multidimensional matrices
- MultidimensionalMatrix _ Sequence -- get entry of multidimensional matrix
- multidimensionalMatrix(RingElement) -- make a multidimensional matrix from a multilinear form
- "permute(MultidimensionalMatrix,List)" -- see permute -- permute the dimensions of a multidimensional matrix
- "randomMultidimensionalMatrix(VisibleList)" -- see randomMultidimensionalMatrix -- random multidimensional matrix
- rank(MultidimensionalMatrix) -- about the border rank of a multidimensional matrix
- "reverseShape(MultidimensionalMatrix)" -- see reverseShape -- reverse the dimensions of a multidimensional matrix
- ring(MultidimensionalMatrix) -- ring of a multidimensional matrix
- RingElement * MultidimensionalMatrix -- product of a scalar with a multidimensional matrix
- "shape(MultidimensionalMatrix)" -- see shape -- shape of a multidimensional matrix
- "sortShape(MultidimensionalMatrix)" -- see sortShape -- sort the dimensions of a multidimensional matrix
- "sparseDiscriminant(Matrix)" -- see sparseDiscriminant -- sparse discriminant (A-discriminant)
- "sparseDiscriminant(RingElement)" -- see sparseDiscriminant -- sparse discriminant (A-discriminant)
- SparseDiscriminant Thing -- evaluate a sparse discriminant
- "sparseResultant(Thing)" -- see sparseResultant -- sparse resultant (A-resultant)
- SparseResultant Thing -- evaluate a sparse resultant
For the programmer
The object SparseResultants is a package.