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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 -- 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 Stratifications

## 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
- PathSignatures -- working with algebraic path signatures
- 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 -- Collections of cells and binomial ideals
- 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
- Tableaux -- constructing skew tableaux
- Triangulations -- triangulations of polyhedra and point sets

## Commutative Algebra

- A1BrouwerDegrees -- for working with A1-Brouwer degree computations and quadratic forms
- 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
- 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
- MatrixFactorizations -- computing with matrix factorizations of different lengths
- 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
- WeilDivisors -- Weil divisors

## Commutative algebra

- NonPrincipalTestIdeals -- singularities of pairs with non-principal ideals

## 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

- CpMackeyFunctors -- homological algebra with Mackey functors for cyclic groups of prime order
- 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
- MatrixFactorizations -- computing with matrix factorizations of different lengths
- 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 -- for working with A1-Brouwer degree computations and quadratic forms
- CpMackeyFunctors -- homological algebra with Mackey functors for cyclic groups of prime order

## 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 classes, Segre classes and the Euler characteristic
- 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

- ExteriorExtensions -- Builds an algebra extending an exterior algebra
- GradedLieAlgebras -- computations in graded Lie algebras
- HighestWeights -- decompose free resolutions and graded modules with a semisimple Lie group action
- HomotopyLieAlgebra -- Homotopy Lie algebra
- LieAlgebraRepresentations -- Lie algebra representations and characters
- 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

## Singularities

- NonPrincipalTestIdeals -- singularities of pairs with non-principal ideals

## System

- Classic -- a classic Macaulay parser
- JSON -- JSON encoding and decoding
- JSONRPC -- JSON-RPC server
- 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

## See also

- packages -- Macaulay2 packages
- The authors of Macaulay2 packages

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_packages.m2:76:0.