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Resultants :: Resultants

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, discriminant, 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

Certification a gold star

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
  • 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
    • "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 Plucker coordinates to Stiefel coordinates
    • "fromPluckerToStiefel(Matrix)" -- see fromPluckerToStiefel -- convert from Plucker coordinates to Stiefel coordinates
    • "fromPluckerToStiefel(RingElement)" -- see fromPluckerToStiefel -- convert from Plucker 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 Plucker coordinates of a linear subspace
    • "plucker(Ideal,ZZ)" -- see plucker -- get the Plucker coordinates of a linear subspace
    • "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

For the programmer

The object Resultants is a package.