packages provided with Macaulay2

Here is a list of the packages that are distributed with Macaulay2. The ones that have been refereed and published are marked with a star.

Algebraic Geometry

• Cremona -- rational maps between projective varieties
• HyperplaneArrangements -- manipulating finite sets of hyperplanes
• IncidenceCorrespondenceCohomology -- Cohomology on the incidence correspondence, bundle of principal parts, and Lefschetz properties
• Jets -- compute jets of various algebraic, geometric and combinatorial objects
• K3Surfaces -- Explicit constructions of K3 surfaces
• NumericalSemigroups -- Compute the Apery set and invariants of a numerical semigroup ring
• PencilsOfQuadrics -- Clifford Algebra of a pencil of quadratic forms
• SpecialFanoFourfolds -- Hodge-special fourfolds
• Varieties -- routines for working with affine and projective varieties and coherent sheaves on them
• WhitneyStratifications -- Compute Whitney Statifications Algebraically & Stratifies Maps

Algebraic Number Theory

• Cyclotomic -- cyclotomic fields
• EllipticCurves -- group law on elliptic curves (and counting points with brute force methods)
• LLLBases -- Lenstra-Lenstra-Lovasz bases

Algebraic Statistics

• AllMarkovBases -- computing all minimal Markov bases of a configuration matrix
• GraphicalModels -- discrete and Gaussian graphical models
• GraphicalModelsMLE -- maximum likelihood estimates for graphical statistical models
• Markov -- Markov ideals arising from Bayesian networks in statistics
• MultigradedImplicitization -- solving implicitization problems using multigradings
• Probability -- basic probability functions
• StatGraphs -- Graphs specific for algebraic statistics

Applied Algebraic Geometry

• AlgebraicSplines -- splines on simplicial complexes, polytopal complexes, and graphs
• IntegerProgramming -- solving integer programs with Gröbner bases
• Oscillators -- code to analyze graph oscillators
• PhylogeneticTrees -- invariants for group-based phylogenetic models
• ReactionNetworks -- reaction networks

Coding Theory

• CodingTheory -- tools for coding theory

Combinatorial Commutative Algebra

• AbstractSimplicialComplexes -- Abstract Simplicial Complexes
• MonomialOrbits -- Orbit representatives of monomial ideals
• PolyominoIdeals -- binomial ideals of collections of cells
• ResolutionsOfStanleyReisnerRings -- Comparing resolutions of Stanley-Reisner rings and computing various systems of parameters
• SimplicialComplexes -- exploring abstract simplicial complexes within commutative algebra
• SimplicialDecomposability -- various decomposability routines for simplicial complexes
• SimplicialPosets -- constructing Stanley simplicial poset rings
• SRdeformations -- deformations of Stanley-Reisner rings and related computations

Combinatorics

• MatrixSchubert -- functions for investigating ASM and matrix Schubert varieties
• NumericalSemigroups -- Compute the Apery set and invariants of a numerical semigroup ring
• Permutations -- functions for working with permutations
• Posets -- partially ordered sets (posets)
• SymmetricPolynomials -- symmetric polynomials
• Triangulations -- triangulations of polyhedra and point sets

Commutative Algebra

• A1BrouwerDegrees -- A1-Brouwer degree computations
• AdjointIdeal -- adjoint ideals of plane curves and related computations
• AInfinity -- AInfinity structures on free resolutions
• AnalyzeSheafOnP1 -- decompose a Sheaf on P1
• BettiCharacters -- finite group characters on free resolutions and graded modules
• BGG -- Bernstein-Gelfand-Gelfand correspondence
• Binomials -- specialized routines for binomial ideals
• BoijSoederberg -- Betti diagram operations useful for investigating the Boij-Soederberg conjectures
• Bruns -- make a 3-generator ideal with an "any" resolution
• CellularResolutions -- cellular resolutions of monomial ideals
• Chordal -- routines that exploit chordal structure
• CompleteIntersectionResolutions -- Analyzing Resolutions over a Complete Intersection
• ConformalBlocks -- for conformal block divisors
• CorrespondenceScrolls -- correspondence scrolls
• Depth -- aids in computations related to depth
• DeterminantalRepresentations -- determinantal representations
• DGAlgebras -- Data type for DG algebras
• DiffAlg -- specialized routines for differential forms
• Divisor -- Weil divisors
• EagonResolution -- Compute the Eagon Resolution of the residue field
• Elimination -- elimination of variables
• EliminationMatrices -- resultants
• EnumerationCurves -- Enumeration of rational curves via torus actions
• ExteriorIdeals -- monomial ideals over exterior algebras
• ExteriorModules -- monomial modules over exterior algebras
• FiniteFittingIdeals -- Fitting ideals of finite modules
• FrobeniusThresholds -- F-thresholds
• FunctionFieldDesingularization -- desingularization of function fields
• GenericInitialIdeal -- find the generic initial ideal of a given ideal
• GeometricDecomposability -- checking whether ideals are geometrically vertex decomposable
• GroebnerStrata -- computing Groebner loci in Hilbert schemes
• Hadamard -- Hadamard products of projective subvarieties
• HigherCIOperators -- higher CI operators
• HodgeIntegrals -- Hodge integrals on the moduli space of curves
• IncidenceCorrespondenceCohomology -- Cohomology on the incidence correspondence, bundle of principal parts, and Lefschetz properties
• IntegralClosure -- integral closure
• InverseSystems -- equivariant Macaulay inverse systems
• Isomorphism -- probabilistic test of isomorphism between modules
• K3Carpets -- K3 double structure on scrolls
• Kronecker -- Kronecker and rational normal forms
• KustinMiller -- unprojection and the Kustin-Miller complex construction
• LexIdeals -- lexicographic-type monomial ideals
• LinearTruncations -- find the multigraded truncations that give linear resolutions
• LocalRings -- operations over a local ring R_p
• M0nbar -- calculations for divisors and F-curves on the moduli space of stable n-pointed genus zero curves
• MatrixSchubert -- functions for investigating ASM and matrix Schubert varieties
• MCMApproximations -- MCM approximations and complete intersections
• MinimalPrimes -- minimal primes and radical routines for ideals
• Miura -- arithmetic of Miura curves
• MixedMultiplicity -- Mixed Multiplicities of ideals
• MonomialAlgebras -- monomial algebras
• MonomialIntegerPrograms -- using integer programming for fast computations with monomial ideals
• MultigradedBGG -- the multigraded BGG correspondence and differential modules
• MultigradedImplicitization -- solving implicitization problems using multigradings
• MultiGradedRationalMap -- degree and birationality of multi-graded rational maps
• MultiplicitySequence -- computing the multiplicity sequence of an ideal
• NoetherianOperators -- algorithms for computing local dual spaces and sets of Noetherian operators
• NoetherNormalization -- place an ideal in Noether normal position
• NonminimalComplexes -- non-minimal strands of a non-minimal resolution of a homogeneous module
• NumericalSemigroups -- Compute the Apery set and invariants of a numerical semigroup ring
• OIGroebnerBases -- OI-modules over Noetherian polynomial OI-algebras
• Parametrization -- rational parametrization of rational plane curves and related computations
• PencilsOfQuadrics -- Clifford Algebra of a pencil of quadratic forms
• Permanents -- permanents of a matrix
• PrimaryDecomposition -- primary decomposition and associated primes routines
• PruneComplex -- pruning chain complexes over polynomial and local rings
• PseudomonomialPrimaryDecomposition -- Primary decomposition of square free pseudomonomial ideals
• Pullback -- pullback of rings
• PushForward -- push forwards of finite ring maps
• QthPower -- An implementation of the Qth-Power algorithm for computing integral closures
• Quasidegrees -- quasidegrees and graded local cohomology
• QuillenSuslin -- the Quillen-Suslin algorithm for bases of projective modules
• RationalMaps -- rational maps between varieties
• RationalPoints -- find the common zeroes of a set of polynomials with coefficients in a finite field
• RationalPoints2 -- find the rational points on a variety
• ReesAlgebra -- Rees algebras
• Regularity -- Castelnuovo-Mumford regularity of a homogeneous ideal
• RelativeCanonicalResolution -- the relative canonical resolution for g-nodal canonical curves with a fixed g^1_k
• ResidualIntersections -- conditions associated to residual intersection theory
• Resultants -- resultants, discriminants, and Chow forms
• SagbiGbDetection -- finding term orders for which the given generators of an ideal/algebra form a Gröbner/SAGBI basis
• Saturation -- quotient, saturation, and annihilator routines for ideals and modules
• SCMAlgebras -- sequentially Cohen-Macaulay modules or ideals
• SectionRing -- the section ring of a Weil Divisor
• Seminormalization -- seminormalization of rings
• SlackIdeals -- Slack ideals of polytopes and matroids
• SparseResultants -- computations with sparse resultants
• StronglyStableIdeals -- studying strongly stable ideals related to Hilbert polynomials
• SubalgebraBases -- Canonical subalgebra bases (aka SAGBI/Khovanskii bases)
• SVDComplexes -- SVD (singular value decomposition) of a complex over the reals and related functions
• SwitchingFields -- Switch Base Fields and Obtain Natural Maps
• SymbolicPowers -- symbolic powers
• TangentCone -- tangent cones
• TateOnProducts -- Tate resolutions on products of projective spaces
• TensorComplexes -- multilinear algebra with labeled bases
• TestIdeals -- singularities in positive characteristic
• TriangularSets -- triangular decompositions of ideals
• Triplets -- triplets of degree sequences and associated Betti diagrams and cohomology tables
• Truncations -- truncation of a module
• TSpreadIdeals -- t-spread ideals of a polynomial ring
• Valuations -- implementation of valuations for rings
• VectorFields -- vector fields
• VirtualResolutions -- Methods for virtual resolutions on products of projective spaces

Complex Analytic Geometry

• EllipticIntegrals -- uniformization of elliptic curves

Convex Geometry

• FourierMotzkin -- convex hulls and polar cones
• LatticePolytopes -- lattice polytopes
• OldPolyhedra -- convex polyhedra
• Polyhedra -- convex polyhedra
• ReflexivePolytopesDB -- simple access to Kreuzer-Skarke database of reflexive polytopes of dimensions 3 and 4
• StatePolytope -- the state polytope of an ideal

D-modules

• BernsteinSato -- Bernstein-Sato
• ConnectionMatrices -- connection matrices and integrable systems from D-ideals
• Dmodules -- D-modules
• HolonomicSystems -- Holonomic Systems
• MultiplierIdeals -- multiplier ideals, log canonical thresholds, and jumping numbers
• MultiplierIdealsDim2 -- multiplier ideals in 2-dimensional rings
• WeylAlgebras -- D-modules

Deformation Theory

• ModuleDeformations -- versal deformations of maximal Cohen-Macaulay modules
• VersalDeformations -- versal deformations and local Hilbert schemes

Documentation

• BeginningMacaulay2 -- Mathematicians' Introduction to Macaulay2
• FirstPackage -- an example Macaulay2 package
• Macaulay2Doc -- Macaulay2 documentation
• PackageTemplate -- an example Macaulay2 package
• SimpleDoc -- a simple documentation function
• Style -- style sheets and images for the documentation
• Text -- documentation and hypertext
• VNumber -- compute v-number of homogeneous ideals and v-function of monomial ideals

Edge Ideals

• BinomialEdgeIdeals -- binomial edge ideals
• EdgeIdeals -- edge ideals

Equivariant Cohomology

• GKMVarieties -- computations with GKM manifolds and moment graphs

Examples and Random Objects

• ExampleSystems -- database of polynomial systems
• Points -- sets of points
• QuadraticIdealExamplesByRoos -- Examples of Quadratic Ideals with Embedding Dimension Four by Jan-Erik Roos
• QuaternaryQuartics -- code to support the paper 'Quaternary Quartic Forms and Gorenstein Rings'
• RandomCanonicalCurves -- Construction of random smooth canonical curves up to genus 14
• RandomComplexes -- random complexes over fields or the integers
• RandomCurves -- random curves
• RandomCurvesOverVerySmallFiniteFields -- general canonical curves of genus <= 15 over fields with small characteristic
• RandomGenus14Curves -- random smooth curves of genus 14
• RandomIdeals -- creating random ideals of various sorts
• RandomMonomialIdeals -- Erdos-Renyi-type random monomial ideals
• RandomObjects -- a framework for making random objects in algebraic geometry
• RandomPlaneCurves -- random plane curves
• RandomPoints -- find a point in a given variety over a finite field
• RandomSpaceCurves -- random smooth space curves
• SpaceCurves -- space curves

Flag Varieties

• GKMVarieties -- computations with GKM manifolds and moment graphs
• IncidenceCorrespondenceCohomology -- Cohomology on the incidence correspondence, bundle of principal parts, and Lefschetz properties
• MatchingFields -- Toric degenerations of flag varieties via matching fields
• NumericalSchubertCalculus -- numerical methods in Schubert Calculus

Game Theory

• GameTheory -- computing equilibria in game theory

Graph Theory

• Chordal -- routines that exploit chordal structure
• GraphicalModels -- discrete and Gaussian graphical models
• Graphs -- graphs and directed graphs (digraphs)
• Nauty -- interface to nauty
• NautyGraphs -- interface to nauty (Graphs fork)
• StatGraphs -- Graphs specific for algebraic statistics

Graphics

• Graphics -- create graphics
• VectorGraphics -- producing SVG graphics
• Visualize -- interactive visualization and manipulation of combinatorial objects in a browser

Groebner Basis Algorithms

• BIBasis -- involutive Pommaret basis in a Boolean ring
• BooleanGB -- Groebner bases for ideals in Boolean polynomial quotient rings
• EquivariantGB -- Equivariant Groebner bases and related algorithms
• FGLM -- Groebner bases via the FGLM algorithm
• GroebnerWalk -- Groebner bases via the Groebner walk
• InvolutiveBases -- Methods for Janet bases and Pommaret bases in Macaulay 2
• Msolve -- interface to the msolve library for solving multivariate polynomial systems using Groebner Bases
• ThreadedGB -- Compute a Groebner basis using the classical Buchberger with multiple threads

Group Theory

• ConwayPolynomials -- a database of Conway polynomials
• FormalGroupLaws -- commutative formal group laws
• InvariantRing -- invariants of group actions

Homological Algebra

• ChainComplexExtras -- some additional ChainComplex Functions
• ChainComplexOperations -- sym2, wedge2, chi2 of a ChainComplex
• Complexes -- beta testing new version of chain complexes
• HighestWeights -- decompose free resolutions and graded modules with a semisimple Lie group action
• Isomorphism -- probabilistic test of isomorphism between modules
• NonminimalComplexes -- non-minimal strands of a non-minimal resolution of a homogeneous module
• OldChainComplexes -- consolidating legacy chain complex code
• PruneComplex -- pruning chain complexes over polynomial and local rings
• ResLengthThree -- Multiplication in free resolutions of length three
• SchurComplexes -- Schur functors of complexes
• SchurFunctors -- Schur modules and maps between them
• SpectralSequences -- spectral sequences
• SVDComplexes -- SVD (singular value decomposition) of a complex over the reals and related functions
• TorAlgebra -- classification of local rings based on multiplication in homology
• Varieties -- routines for working with affine and projective varieties and coherent sheaves on them
• VirtualResolutions -- Methods for virtual resolutions on products of projective spaces

Homotopy Theory

• A1BrouwerDegrees -- A1-Brouwer degree computations

Interfaces

• Bertini -- interface to Bertini
• CohomCalg -- interface to cohomCalg software for computing sheaf cohomology of line bundles on toric varieties
• CoincidentRootLoci -- coincident root loci
• ConvexInterface -- interface to Convex
• ForeignFunctions -- foreign function interface
• FourTiTwo -- Interface to 4ti2
• gfanInterface -- interface to Anders Jensen's Gfan software
• MapleInterface -- interface to Maple
• Msolve -- interface to the msolve library for solving multivariate polynomial systems using Groebner Bases
• Nauty -- interface to nauty
• NautyGraphs -- interface to nauty (Graphs fork)
• NCAlgebra -- data types for noncommutative algebras
• Normaliz -- interface to Normaliz in Macaulay2
• PHCpack -- interface to PHCpack
• Polymake -- interfacing with polymake
• Python -- interface to Python
• RInterface -- interface to R for statistical computing
• SemidefiniteProgramming -- semidefinite programming
• StatePolytope -- the state polytope of an ideal
• Topcom -- interface to the topcom software package which in particular computes triangulations

Intersection Theory

• AbstractToricVarieties -- links abstract simplicial (normal) toric varieties to Schubert2
• Book3264Examples -- examples to accompany the eponymous book by Eisenbud and Harris
• CharacteristicClasses -- CSM and Segre classes and the Euler characteristic for subschemes of smooth complete toric varieties
• CotangentSchubert -- Cotangent Schubert calculus
• Schubert2 -- characteristic classes for varieties without equations
• SegreClasses -- Segre class computations for containment of varieties and Fulton-MacPherson intersection products

Lie Groups and Lie Algebras

• GradedLieAlgebras -- computations in graded Lie algebras
• HighestWeights -- decompose free resolutions and graded modules with a semisimple Lie group action
• HomotopyLieAlgebra -- Homotopy Lie algebra
• LieTypes -- common types and methods for Lie groups and Lie algebras
• WeylGroups -- root systems and Weyl groups

Linear Algebra

• FastMinors -- faster linear algebra operations
• LLLBases -- Lenstra-Lenstra-Lovasz bases
• SuperLinearAlgebra -- computations related to supermatrices

Matroids

• GKMVarieties -- computations with GKM manifolds and moment graphs
• HyperplaneArrangements -- manipulating finite sets of hyperplanes
• Matroids -- computations with matroids
• SlackIdeals -- Slack ideals of polytopes and matroids

Miscellaneous

• Benchmark -- standard Macaulay2 benchmarks
• Browse -- a method for browsing and examining Macaulay2 data structures
• EngineTests -- a test suite for the Macaulay2 engine
• PackageCitations -- citation of Macaulay2 packages
• Units -- units conversion and physical constants
• XML -- an XML parser

Noncommutative Algebra

• AssociativeAlgebras -- Noncommutative algebra
• NCAlgebra -- data types for noncommutative algebras

Numerical Algebraic Geometry

• Bertini -- interface to Bertini
• DecomposableSparseSystems -- Solving decomposable sparse systems
• EigenSolver -- polynomial system solver via eigen-computations
• EllipticIntegrals -- uniformization of elliptic curves
• MonodromySolver -- solving polynomial systems via monodromy
• NAGtypes -- types used in Numerical Algebraic Geometry
• NoetherianOperators -- algorithms for computing local dual spaces and sets of Noetherian operators
• NumericalAlgebraicGeometry -- numerical algebraic geometry
• NumericalCertification -- numerical certification
• NumericalImplicitization -- numerical invariants of images of varieties
• NumericalSchubertCalculus -- numerical methods in Schubert Calculus
• NumericSolutions -- solving systems of polynomial equalities
• PHCpack -- interface to PHCpack
• SLPexpressions -- straight line programs and algebraic circuits

Numerical Linear Algebra

• NumericalLinearAlgebra -- numerically compute local dual space and Hilbert functions

Projective Algebraic Geometry

• AdjunctionForSurfaces -- Adjunction for Surfaces
• Brackets -- Brackets, Grassmann-Cayley Algebra, and Projective Geometry
• MultiprojectiveVarieties -- multi-projective varieties and multi-rational maps
• PlaneCurveLinearSeries -- Linear series on the normalization of a plane curve
• SchurVeronese -- Data for Veronese embeddings of projective space
• TerraciniLoci -- Terracini loci of projective varieties

Real Algebraic Geometry

• CoincidentRootLoci -- coincident root loci
• RealRoots -- symbolically exploring, counting, and locating real solutions to general polynomial systems
• SemidefiniteProgramming -- semidefinite programming
• SumsOfSquares -- sums of squares

Representation Theory

• DeterminantalRepresentations -- determinantal representations
• InvariantRing -- invariants of group actions
• PieriMaps -- maps between representations of the general linear group based on the Pieri formulas
• SchurComplexes -- Schur functors of complexes
• SchurFunctors -- Schur modules and maps between them
• SchurRings -- representation rings of general linear groups and of symmetric groups
• SLnEquivariantMatrices -- file ancillary to the paper "A construction of equivariant bundles on the space of symmetric forms"
• SpechtModule -- invariants for permutation groups
• SymmetricPolynomials -- symmetric polynomials

System

• Classic -- a classic Macaulay parser
• JSON -- JSON encoding and decoding
• MergeTeX -- Run Macaulay2 code inside a LaTeX file
• OnlineLookup -- Look up mathematical information online
• OpenMath -- OpenMath for Macaulay2
• Parsing -- a framework for creating recursive descent parsers
• RunExternalM2 -- run Macaulay2 functions outside the current Macaulay2 process
• SCSCP -- SCSCP for Macaulay2
• Serialization -- reversible conversion of all Macaulay2 objects to strings

Toric Geometry

• AbstractToricVarieties -- links abstract simplicial (normal) toric varieties to Schubert2
• CohomCalg -- interface to cohomCalg software for computing sheaf cohomology of line bundles on toric varieties
• GKMVarieties -- computations with GKM manifolds and moment graphs
• NormalToricVarieties -- routines for working with normal toric varieties and related objects
• OldToricVectorBundles -- vector bundles on toric varieties
• PositivityToricBundles -- check positivity of toric vector bundles
• ThinSincereQuivers -- Construction of flow polytopes and their associated quivers
• ToricHigherDirectImages -- computations involving pushforwards and higher direct images of toric maps
• ToricInvariants -- Euclidean distance degrees, polar degrees, and Chern-Mather classes of toric varieties
• ToricTopology -- toric topology
• ToricVectorBundles -- vector bundles on toric varieties

Tropical Geometry

• Tropical -- computations in tropical geometry
• TropicalToric -- tropical methods for toric intersection theory