Rings: efficient Java/Scala library for polynomial rings

Rings is an efficient lightweight library for commutative algebra. Polynomial arithmetic, GCDs, polynomial factorization and Groebner bases are implemented with the use of modern asymptotically fast algorithms. Rings can be easily interacted or embedded in applications via simple API with fully typed hierarchy of algebraic structures and algorithms for commutative algebra. As well, an interactive REPL is also provided. The use of Scala language brings a quite novel powerful strongly typed functional programming model allowing to write short, expressive and fast code for applications. At the same time Rings shows one of the best or even unmatched in some cases performance among existing software for algebraic calculations.

The key features of Rings include:

• Rings →
  integers, fractions, finite and algebraic fields, multiple field extensions, polynomial rings and more
• Polynomials →
  Efficient univariate and multivariate polynomials over arbitrary coefficient rings
• Polynomial GCD →
  Highly performant polynomial GCD over arbitrary coefficient domains
• Univariate and multivariate polynomial factorization →
  Highly performant polynomial factorization over almost arbitrary rings
• Ideals and Gröbner bases →
  Polynomial ideals and efficient algorithms for Gröbner bases
• Scala DSL →
  Powerful domain specific language in Scala
• Fast →
  Really fast library suitable for real-world computational challenges

For a quick overview of what Rings can do proceed to Quick Tour and try out Rings.repl.

Rings sources are hosted at GitHub: https://github.com/PoslavskySV/rings.

Documentation:

• Quick Tour
  • Set up
    • Interactive Rings shell
    • Java/Scala library
  • Examples: rings, ideals, Gröbner bases, GCDs & factorization
    • Some built-in rings
    • Univariate polynomials
    • Multivariate polynomials
    • Rational function arithmetic
    • Ideals and Gröbner bases
    • Programming
  • Highlighted benchmarks
• User guide
  • Rings library structure
  • Numbers
    • Integers
    • Prime numbers
    • Modular arithmetic with machine integers
  • Rings
    • List of built-in rings
    • Galois fields
    • Algebraic number fields and field extensions
    • Fields of fractions
    • Rational function arithmetic
    • Univariate polynomial rings
    • Multivariate polynomial rings
    • Quotient rings
  • Scala DSL
    • General mathematical operators
    • Polynomial operators
    • Univariate polynomial operators
    • Multivariate polynomial operators
    • Ring methods
    • Polynomial ring methods
    • Ideal methods
  • Input/Output
    • Java
    • Scala
  • Polynomials
    • String representation of polynomials
    • Polynomial instances and mutability
    • Polynomial GCD, factorization and division with remainder
    • Univariate polynomials
    • Multivariate polynomials
  • Ideals in multivariate polynomial rings
    • Hilbert-Poincare series
    • Gröbner bases algorithms
• Benchmarks
  • Multivariate GCD
    • Multivariate GCD over \(Z_2\)
    • Multivariate GCD over \(Z\)
  • Multivariate factorization
    • Multivariate factorization over \(Z_2\)
    • Multivariate factorization over \(Z_{524287}\)
    • Multivariate factorization over \(Z\)
    • Multivariate factorization on large not very sparse polynomials
  • Univariate factorization
• Index of algorithms implemented in Rings
  • References

Citation

Please, cite this paper if you use Rings:

[POS19]

Stanislav Poslavsky, Rings: An efficient Java/Scala library for polynomial rings, Computer Physics Communications, Volume 235, 2019, Pages 400-413, doi:10.1016/j.cpc.2018.09.005

License

Apache License, Version 2.0 http://www.apache.org/licenses/LICENSE-2.0.txt