Promote integer to rational for division

Author: blegat

docs/src/division.md

@@ -8,12 +8,19 @@ Given a polynomial ``p`` and divisors ``d_1, \ldots, d_n``, one can find ``r`` a
 You can obtain the vector ``[q_1, \ldots, q_n]`` using `div(p, d)` where ``d = [d_1, \ldots, d_n]`` and ``r`` using the `rem` function with the same arguments.
 The `divrem` function returns ``(q, r)``.
 ```@docs
+divrem
+div
+rem
 divides
 div_multiple
 ```
 
 Note that the coefficients of the polynomials need to be a field for `div`,
 `rem` and `divrem` to work.
+If the coefficient type is not a field, it is promoted to a field using [`promote_to_field`](@ref).
+```@docs
+promote_to_field
+```
 Alternatively, [`pseudo_rem`](@ref) or [`pseudo_divrem`](@ref) can be used
 instead as they do not require the coefficient type to be a field.
 ```@docs
@@ -30,6 +37,7 @@ The same functions can be used with monomials and polynomials:
 ```@docs
 gcd
 lcm
+<<<<<<< HEAD
 AbstractUnivariateGCDAlgorithm
 GeneralizedEuclideanAlgorithm
 SubresultantAlgorithm
@@ -42,4 +50,6 @@ univariate_gcd
 content
 primitive_part
 primitive_part_content
+=======
+>>>>>>> 1fe1303 (Promote integer to rational for division)
 ```

src/division.jl

@@ -64,8 +64,25 @@ struct Field end
 struct UniqueFactorizationDomain end
 const UFD = UniqueFactorizationDomain
 
+"""
+    promote_to_field(::Type{T})
+
+Promote the type `T` to a field. For instance, `promote_to_field(T)` returns
+`Rational{T}` if `T` is an integer and `promote_to_field(T)` returns `RationalPoly{T}`
+if `T` is a polynomial.
+"""
+function promote_to_field end
+
+function promote_to_field(::Type{T}) where {T<:Integer}
+    return Rational{T}
+end
+function promote_to_field(::Type{T}) where {T<:_APL}
+    return RationalPoly{T,T}
+end
+promote_to_field(::Type{T}) where {T} = T
+
 algebraic_structure(::Type{<:Integer}) = UFD()
-algebraic_structure(::Type{<:AbstractPolynomialLike}) = UFD()
+algebraic_structure(::Type{<:_APL}) = UFD()
 # `Rational`, `AbstractFloat`, JuMP expressions, etc... are fields
 algebraic_structure(::Type) = Field()
 _field_absorb(::UFD, ::UFD) = UFD()
@@ -430,7 +447,11 @@ function MA.promote_operation(
     ::Type{P},
     ::Type{Q},
 ) where {T,S,P<:_APL{T},Q<:_APL{S}}
-    U = MA.promote_operation(/, T, S)
+    U = MA.promote_operation(
+        /,
+        promote_to_field(T),
+        promote_to_field(S),
+    )
     # `promote_type(P, Q)` is needed for TypedPolynomials in case they use different variables
     return polynomial_type(promote_type(P, Q), MA.promote_operation(-, U, U))
 end

test/division.jl

@@ -54,8 +54,14 @@ end
 
 function divrem_test()
     Mod.@polyvar x y
-    @test (@inferred div(x * y^2 + 1, x * y + 1)) == y
-    @test (@inferred rem(x * y^2 + 1, x * y + 1)) == -y + 1
+    p = x * y^2 + 1
+    q = x * y + 1
+    @test typeof(div(p, q)) == MA.promote_operation(div, typeof(p), typeof(q))
+    @test typeof(rem(p, q)) == MA.promote_operation(rem, typeof(p), typeof(q))
+    @test coefficient_type(div(p, q)) == Rational{Int}
+    @test coefficient_type(rem(p, q)) == Rational{Int}
+    @test (@inferred div(p, q)) == y
+    @test (@inferred rem(p, q)) == -y + 1
     @test (@inferred div(x * y^2 + x, y)) == x * y
     @test (@inferred rem(x * y^2 + x, y)) == x
     @test (@inferred rem(x^4 + x^3 + (1 + 1e-10) * x^2 + 1, x^2 + x + 1)) == 1