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Getting Started

Singular.jl is a Julia interface to the Singular computer algebra system. It was written by Oleksandr Motsak, William Hart and other contributors, and is maintained by Hans Schoenemann and Max Horn. It is part of the Oscar project which is supported by the Deutsche Forschungsgemeinschaft DFG within the Collaborative Research Center TRR 195.

The features of Singular so far include:

  • Singular integers, rationals Z/nZ, Z/pZ, Galois fields
  • Multivariate polynomials, including several noncommutative variants
  • Ideals over polynomial rings
  • Free modules over polynomial rings and submodules given by a finite generating set
  • Groebner basis over a field
  • Free/minimal resolutions
  • Syzygy modules
  • Nemo.jl rings can be used as coefficient rings

Installation

To use Singular.jl we require Julia 1.6 or higher. Please see https://julialang.org/downloads/ for instructions on how to obtain julia for your system.

At the Julia prompt simply type

julia> using Pkg; Pkg.add("Singular")

Here is an example of using Singular.jl

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
(Singular polynomial ring (QQ),(x,y),(dp(2),C), spoly{n_Q}[x, y])

julia> I = Ideal(R, x^2 + 1, x*y + 1)
Singular ideal over Singular polynomial ring (QQ),(x,y),(dp(2),C) with generators (x^2 + 1, x*y + 1)

julia> G = std(I)
Singular ideal over Singular polynomial ring (QQ),(x,y),(dp(2),C) with generators (x - y, y^2 + 1)

julia> Z = syz(G)
Singular module over Singular polynomial ring (QQ),(x,y),(dp(2),C), with generators:
y^2*gen(1)-x*gen(2)+y*gen(2)+gen(1)

julia> F = fres(G, 0)
Singular resolution: R^1 <- R^2 <- R^1

julia> F[1]
Singular ideal over Singular polynomial ring (QQ),(x,y),(dp(2),C) with generators (x - y, y^2 + 1)