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 the preprint Coisotropic Hypersurfaces in the Grassmannian, by K. Kohn.

- Giovanni Staglianò <giovannistagliano@gmail.com>

Version **1.2.1** of this package was accepted for publication in volume 8 of the journal 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, https://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/Resultants.m2, commit number 61c93a6aaf9d6bf0dd11440339145703ce3d824b.

- 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
- discriminant -- resultant of the partial derivatives
- dualize -- apply duality of Grassmannians
- dualVariety -- projective dual variety
- fromPluckerToStiefel -- convert from Plucker 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 Plucker coordinates of a linear subspace
- resultant -- multipolynomial resultant
- tangentialChowForm -- higher Chow forms of a projective variety
- 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 Plucker coordinates
- SingularLocus -- pass the singular locus of the variety