# 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

• Giovanni Staglianò

## 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
• 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 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
• "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 .