Rational field

Singular.jl provides rational numbers via Singular's n_Q type.

There is a constant parent object representing the field of rationals, called QQ in Singular.jl. It is defined by QQ = Rationals(), which calls the constructor for the unique field of rationals in Singular.

Rational functionality

The rationals in Singular.jl implement the Field interface defined by AbstractAlgebra.jl. They also implement the Fraction Field interface.

https://nemocas.github.io/AbstractAlgebra.jl/fields.html

https://nemocas.github.io/AbstractAlgebra.jl/fraction_fields.html

We describe here only the extra functionality provided by Singular that is not already described in those interfaces.

Constructors

In addition to the standard constructors required for the interfaces listed above, Singular.jl provides the following constructors.

QQ(n::n_Z)
QQ(n::fmpz)

Construct a Singular rational from the given integer $n$.

Basic manipulation

Base.numerator — Method.

numerator(n::n_Q)

Return the numerator of the given fraction.

source

Base.denominator — Method.

denominator(n::n_Q)

Return the denominator of the given fraction.

source

Base.abs — Method.

abs(n::n_Q)

Return the absolute value of the given fraction.

source

f = QQ(-12, 7)

h = numerator(QQ)
k = denominator(QQ)
m = abs(f)

Comparison

Here is a list of the comparison functions implemented, with the understanding that isless provides all the usual comparison operators.

Function
`isless(a::n_Q, b::n_Q)`

We also provide the following ad hoc comparisons which again provide all of the comparison operators mentioned above.

Function
`isless(a::n_Q, b::Integer)`
`isless(a::Integer, b::n_Q)`

Examples

a = QQ(12, 7)
b = QQ(-3, 5)

a > b
a != b
a > 1
5 >= b

Rational reconstruction

Nemo.reconstruct — Method.

reconstruct(x::n_Z, y::n_Z)

Given $x$ modulo $y$, find $r/s$ such that $x \equiv r/s \pmod{y}$ for values $r$ and $s$ satisfying the bound $y > 2(|r| + 1)(s + 1)$.

source

The following ad hoc versions of the same function also exist.

reconstruct(::n_Z, ::Integer)
reconstruct(::Integer, ::n_Z)

Examples

q1 = reconstruct(ZZ(7), ZZ(3))
q2 = reconstruct(ZZ(7), 5)