All rational functions definend over a ring lie in the same family, the rational functions family over this ring.
In GAP therefore the family of a polynomial depends only on the family of the coefficients, all polynomials whose coefficients lie in the same family are ``compatible''.
RationalFunctionsFamily( fam ) A
creates a family containing rational functions with coefficients
in fam. This family fam must be a UFD, that is to say, there are no
zero divisors, the family must have a one, be comutative and the
factorisation of an elements into irreducible elements of the family
must be unique (up to units and order).
All elements of the RationalFunctionsFamily are rational functions
(see IsRationalFunction).
IsRationalFunctionsFamily( obj ) C
Is the category of a family of rational functions.
gap> fam:=RationalFunctionsFamily(FamilyObj(1)); NewFamily( "RationalFunctionsFamily(...)", [ 618, 620 ], [ 82, 85, 89, 93, 97, 100, 103, 107, 111, 618, 620 ] )
CoefficientsFamily( rffam ) A
If rffam has been created as RationalFunctionsFamily(cfam) this
attribute holds the coefficients family cfam.
GAP does not embed the base ring in the polynomial ring. While multiplication and addition of base ring elements to rational functions return the expected results, polynomials and rational functions are not equal.
gap> 1=Indeterminate(Rationals)^0; false
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GAP 4 manual