Jets -- compute jets of various algebraic, geometric and combinatorial objects

Description

This package enables computations with jet functors. It introduces the jets method to compute jets of polynomial rings, ideals, quotients, ring homomorphisms, and varieties. The construction of jets follows an algebraic procedure discussed in several sources, including the first three references below.

Additional features include an alternative algorithm to compute the radical of jets of monomial ideals, a function to construct jets of graphs, a method for principal components of jets, and an option for jets with "projective" gradings.

Version history:

1.1: JSAG version.
1.2: Improved method for principal components. Added methods for invariants of principal components of monomial ideals.

References

L. Ein and M. Mustaţă, Jet schemes and singularities.

P. Vojta, Jets via Hasse-Schmidt Derivations.

R.A. Goward and K.E. Smith, The jet scheme of a monomial scheme.

F. Galetto, E. Helmick, and M. Walsh, Jet graphs.

F. Galetto, N. Iammarino, and T. Yu, Jets and principal components of monomial ideals, and very well-covered graphs

Authors

Federico Galetto <[email protected]>
Nicholas Iammarino <[email protected]>

Certification

Version 1.1 of this package was accepted for publication in volume 12 of The Journal of Software for Algebra and Geometry on 20 October 2022, in the article Computing with jets (DOI: 10.2140/jsag.2022.12.43). That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 1.2 of Jets.

Source code

The source code from which this documentation is derived is in the file Jets.m2. The auxiliary files accompanying it are in the directory Jets/.

Exports

Functions and commands
- jets -- compute the jets of an object
- jetsProjection -- canonical map between jets rings
- jetsRadical -- compute radicals of jets ideals
- liftingFunction -- compute values of a lifting function
- liftingMatrix -- arrange values of lifting function in a matrix
- principalComponent -- compute principal component of jets
Methods
- jets(ZZ,AffineVariety) -- the jets of an affine variety
- jets(ZZ,Graph) -- the jets of a graph
- jets(ZZ,HyperGraph) -- see jets(ZZ,Graph) -- the jets of a graph
- jets(ZZ,Ideal) -- compute jets of a an ideal in a polynomial ring
- jets(ZZ,PolynomialRing) -- compute jets of a polynomial ring
- jets(ZZ,QuotientRing) -- the jets of an affine algebra
- jets(ZZ,RingMap) -- the jets of a homomorphism of rings
- jetsProjection(ZZ,ZZ,PolynomialRing) -- see jetsProjection -- canonical map between jets rings
- jetsProjection(ZZ,ZZ,QuotientRing) -- see jetsProjection -- canonical map between jets rings
- jetsRadical(ZZ,Ideal) -- see jetsRadical -- compute radicals of jets ideals
- liftingFunction(ZZ,ZZ,ZZ) -- see liftingFunction -- compute values of a lifting function
- liftingMatrix(ZZ,ZZ,ZZ) -- see liftingMatrix -- arrange values of lifting function in a matrix
- principalComponent(ZZ,Ideal) -- see principalComponent -- compute principal component of jets
Symbols
- jet -- hashtable key
- jetsBase -- hashtable value
- jetsInfo -- hashtable key
- jetsMatrix -- hashtable value
- jetsMaxOrder -- hashtable value
- jetsRing -- hashtable value
- projet -- hashtable key
- Saturate -- option for principal components
Other things
- JJ -- scripted functor associated with jets

For the programmer

The object Jets is a package.