types.md

CurrentModule = AlgebraicSolving
DocTestSetup = quote
  using AlgebraicSolving
end

using AlgebraicSolving

Pages = ["types.md"]

Data Types

Introduction

AlgebraicSolving handles ideals in multivariate polynomial rings over a prime field of characteristic $0$ or $p$ where $p$ is a prime number $<2^{31}$.

Polynomial Rings

We use Nemo's multivariate polynomial ring structures:

using AlgebraicSolving
R, (x,y,z) = polynomial_ring(QQ, ["x", "y", "z"], ordering=:degrevlex)

The above example defines a multivariate polynomial ring in three variables x, y, and z over the rationals using the dgree reverse lexicographical ordering for printing polynomials in the following. One can also define polynomial rings over finite fields:

using AlgebraicSolving
R, (x,y,z) = polynomial_ring(GF(101), ["x", "y", "z"], ordering=:degrevlex)

Ideals

Ideals can be constructed by giving an array of generators. Ideals cache varies data structures connected to ideals in order to make computational algebra more effective:

using AlgebraicSolving
R, (x,y,z) = polynomial_ring(QQ, ["x", "y", "z"], ordering=:degrevlex)
I = Ideal([x+y+1, y*z^2-13*y^2])