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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 a gold star -- rational maps between projective varieties
  • HyperplaneArrangements -- manipulating finite sets of hyperplanes
  • Jets a gold star -- compute jets of various algebraic, geometric and combinatorial objects
  • K3Surfaces -- Explicit constructions of K3 surfaces
  • 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

  • GraphicalModels a gold star -- discrete and Gaussian graphical models
  • GraphicalModelsMLE a gold star -- maximum likelihood estimates for graphical statistical models
  • Markov -- Markov ideals arising from Bayesian networks in statistics
  • Probability a gold star -- basic probability functions
  • StatGraphs -- Graphs specific for algebraic statistics

Applied Algebraic Geometry

Coding Theory

Combinatorial Commutative Algebra


Commutative Algebra

Complex Analytic Geometry

Convex Geometry


Deformation Theory


  • 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

Equivariant Cohomology

  • GKMVarieties -- computations with GKM manifolds and moment graphs

Examples and Random Objects

Flag Varieties

Graph Theory

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


  • Graphics -- create graphics
  • VectorGraphics -- A package to produce 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
  • ThreadedGB a gold star -- Compute a Groebner basis using the classical Buchberger with multiple threads

Group Theory

Homological Algebra

  • ChainComplexExtras -- some additional ChainComplex Functions
  • ChainComplexOperations -- sym2, wedge2, chi2 of a ChainComplex
  • Complexes -- development package for beta testing new version of chain complexes
  • HighestWeights a gold star -- 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
  • PruneComplex -- pruning chain complexes over polynomial and local rings
  • ResLengthThree -- Multiplication in free resolutions of length three
  • SchurComplexes a gold star -- 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 a gold star -- classification of local rings based on multiplication in homology
  • Varieties -- routines for working with affine and projective varieties and coherent sheaves on them
  • VirtualResolutions a gold star -- Methods for virtual resolutions on products of projective spaces


Intersection Theory

  • AbstractToricVarieties -- links abstract simplicial (normal) toric varieties to Schubert2
  • Book3264Examples -- examples to accompany the eponymous book by Eisenbud and Harris
  • CharacteristicClasses a gold star -- CSM classes, Segre classes and the Euler characteristic for some subschemes of smooth complete toric varieties
  • CotangentSchubert a gold star -- Cotangent Schubert calculus
  • Schubert2 -- characteristic classes for varieties without equations
  • SegreClasses -- test containment of varieties and computes algebraic multiplicity of subvarieties and Fulton-MacPherson intersection products, via a very general Segre class computation

Lie Groups and Lie Algebras

Linear Algebra



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

Noncommutative Algebra

Numerical Algebraic Geometry

Numerical Linear Algebra

Projective Algebraic Geometry

Real Algebraic Geometry

Representation Theory


  • 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 cohomology of torus invariant divisors on a toric variety
  • GKMVarieties -- computations with GKM manifolds and moment graphs
  • NormalToricVarieties -- routines for working with normal toric varieties and related objects
  • OldToricVectorBundles a gold star -- vector bundles on toric varieties
  • PositivityToricBundles -- check positivity of toric vector bundles
  • ThinSincereQuivers -- Construction of flow polytopes and their associated quivers
  • ToricInvariants -- Euclidean distance degrees, polar degrees, degree and codimension of the dual, and Chern-Mather classes of toric varieties X_A from the polytope conv(A) or from its Gale dual
  • ToricTopology -- toric topology
  • ToricVectorBundles a gold star -- vector bundles on toric varieties

Tropical Geometry

  • Tropical -- A package for doing computations in tropical geometry
  • TropicalToric a gold star -- A package on tropical methods for toric intersection theory

See also