Resultants -- resultants, discriminants, and Chow forms

Description

This package provides methods to deal with resultants and discriminants of multivariate polynomials, and with higher associated subvarieties of irreducible projective varieties. The main methods are: resultant(Matrix), discriminant(RingElement), chowForm, dualVariety, and tangentialChowForm. For the mathematical theory, we refer to the following two books: Using Algebraic Geometry, by David A. Cox, John Little, Donal O'shea; Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky. Other references for the theory of Chow forms are: The equations defining Chow varieties, by M. L. Green and I. Morrison; Multiplicative properties of projectively dual varieties, by J. Weyman and A. Zelevinsky; and Coisotropic hypersurfaces in Grassmannians, by K. Kohn.

Author

Giovanni Staglianò <[email protected]>

Certification

Version 1.2.1 of this package was accepted for publication in volume 8 of The Journal of Software for Algebra and Geometry on 18 May 2018, in the article A package for computations with classical resultants. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 1.2.2 of Resultants.

Source code

The source code from which this documentation is derived is in the file Resultants.m2.

Exports

Functions and commands
- affineDiscriminant -- affine discriminant
- affineResultant -- affine resultant
- cayleyTrick -- Cayley trick
- chowEquations -- Chow equations of a projective variety
- chowForm -- Chow form of a projective variety
- conormalVariety -- conormal variety
- dualize -- apply duality of Grassmannians
- dualVariety -- projective dual variety
- fromPluckerToStiefel -- convert from Plücker coordinates to Stiefel coordinates
- genericPolynomials -- generic homogeneous polynomials
- Grass -- coordinate ring of a Grassmannian
- hurwitzForm -- Hurwitz form of a projective variety
- isCoisotropic -- whether a hypersurface of a Grassmannian is a tangential Chow form
- isInCoisotropic -- test membership in a coisotropic hypersurface
- macaulayFormula -- Macaulay formula for the resultant
- plucker -- get the Plücker coordinates of a linear subspace
- tangentialChowForm -- higher Chow forms of a projective variety
- veronese -- Veronese embedding
Methods
- affineDiscriminant(RingElement) -- see affineDiscriminant -- affine discriminant
- affineResultant(List) -- see affineResultant -- affine resultant
- affineResultant(Matrix) -- see affineResultant -- affine resultant
- cayleyTrick(Ideal,ZZ) -- see cayleyTrick -- Cayley trick
- chowEquations(RingElement) -- see chowEquations -- Chow equations of a projective variety
- chowForm(Ideal) -- see chowForm -- Chow form of a projective variety
- chowForm(RingMap) -- see chowForm -- Chow form of a projective variety
- conormalVariety(Ideal) -- see conormalVariety -- conormal variety
- discriminant(RingElement) -- resultant of the partial derivatives
- dualize(Ideal) -- see dualize -- apply duality of Grassmannians
- dualize(Matrix) -- see dualize -- apply duality of Grassmannians
- dualize(Ring) -- see dualize -- apply duality of Grassmannians
- dualize(RingElement) -- see dualize -- apply duality of Grassmannians
- dualize(RingMap) -- see dualize -- apply duality of Grassmannians
- dualize(VisibleList) -- see dualize -- apply duality of Grassmannians
- dualVariety(Ideal) -- see dualVariety -- projective dual variety
- dualVariety(RingMap) -- see dualVariety -- projective dual variety
- fromPluckerToStiefel(Ideal) -- see fromPluckerToStiefel -- convert from Plücker coordinates to Stiefel coordinates
- fromPluckerToStiefel(Matrix) -- see fromPluckerToStiefel -- convert from Plücker coordinates to Stiefel coordinates
- fromPluckerToStiefel(RingElement) -- see fromPluckerToStiefel -- convert from Plücker coordinates to Stiefel coordinates
- genericPolynomials(List) -- see genericPolynomials -- generic homogeneous polynomials
- genericPolynomials(VisibleList,Ring) -- see genericPolynomials -- generic homogeneous polynomials
- Grass(ZZ,ZZ) -- see Grass -- coordinate ring of a Grassmannian
- Grass(ZZ,ZZ,Ring) -- see Grass -- coordinate ring of a Grassmannian
- hurwitzForm(Ideal) -- see hurwitzForm -- Hurwitz form of a projective variety
- isCoisotropic(RingElement) -- see isCoisotropic -- whether a hypersurface of a Grassmannian is a tangential Chow form
- isInCoisotropic(Ideal,Ideal) -- see isInCoisotropic -- test membership in a coisotropic hypersurface
- macaulayFormula(List) -- see macaulayFormula -- Macaulay formula for the resultant
- macaulayFormula(Matrix) -- see macaulayFormula -- Macaulay formula for the resultant
- plucker(Ideal) -- see plucker -- get the Plücker coordinates of a linear subspace
- plucker(Ideal,ZZ) -- see plucker -- get the Plücker coordinates of a linear subspace
- resultant(List) -- see resultant(Matrix) -- multipolynomial resultant
- resultant(Matrix) -- multipolynomial resultant
- tangentialChowForm(Ideal,ZZ) -- see tangentialChowForm -- higher Chow forms of a projective variety
- veronese(ZZ,ZZ) -- see veronese -- Veronese embedding
- veronese(ZZ,ZZ,Ring) -- see veronese -- Veronese embedding
Symbols
- AffineChartGrass -- use an affine chart on the Grassmannian
- AffineChartProj -- use an affine chart on the projective space
- AssumeOrdinary -- whether the expected codimension is 1
- Duality -- whether to use dual Plücker coordinates
- SingularLocus -- pass the singular locus of the variety

For the programmer

The object Resultants is a package.