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Subsections
+ a : FldFunRatElt -> FldFunRatElt
- a : FldFunRatElt -> FldFunRatElt
a + b : FldFunRatElt, FldFunRatElt -> FldFunRatElt
a - b : FldFunRatElt, FldFunRatElt -> FldFunRatElt
a * b : FldFunRatElt, FldFunRatElt -> FldFunRatElt
a / b : FldFunRatElt, FldFunRatElt -> FldFunRatElt
a ^ k : FldFunRatElt, RngIntElt -> FldFunRatElt
a eq b : FldFunRatElt, FldFunRatElt -> BoolElt
a ne b : FldFunRatElt, FldFunRatElt -> BoolElt
a in F : FldFunRatElt, FldFunRat -> BoolElt
a notin F : FldFunRatElt, FldFunRat -> BoolElt
Given a rational function f∈K, the field of fractions of R,
return the numerator P of f=P/Q as an element
of the polynomial ring R.
Given a rational function f∈K, the field of fractions of R,
return the denominator Q of f=P/Q as an element
of the polynomial ring R.
Given a rational function f in a univariate function field, return
the degree of f as an integer (the maximum of the degree of the numerator of f
and the degree of the denominator of f).
Given a rational function f in a multivariate function field, return
the total degree of f as an integer (the total degree of the numerator of f
minus the total degree of the denominator of f).
Given a rational function f in a multivariate function field, return
the weighted degree of f as an integer (the weighted degree of the
numerator of f minus the weighted degree of the denominator of f).
IsZero(a) : FldFunRatElt -> BoolElt
IsOne(a) : FldFunRatElt -> BoolElt
IsMinusOne(a) : FldFunRatElt -> BoolElt
IsNilpotent(a) : FldFunRatElt -> BoolElt
IsIdempotent(a) : FldFunRatElt -> BoolElt
IsUnit(a) : FldFunRatElt -> BoolElt
IsZeroDivisor(a) : FldFunRatElt -> BoolElt
IsRegular(a) : FldFunRatElt -> BoolElt
Given a univariate rational function f in F, return the rational
function in F obtained by evaluating the
indeterminate in r, which must be from (or coercible into)
the coefficient ring of the integers of F.
Given a multivariate rational function f in F, return the rational
function in F obtained by evaluating the v-th variable
in r, which must be from (or coercible into)
the coefficient ring of the integers of F.
Given a univariate rational function f, return the first
derivative of f with respect to its variable.
Given a univariate rational function f, return the k-th
derivative of f with respect to its variable. k must be non-negative.
Given a multivariate rational function f, return the first
derivative of f with respect to variable number v.
Given a multivariate rational function f, return the k-th
derivative of f with respect to variable number v.
k must be non-negative.
Given a univariate rational function f in F=K(x), return the (unique)
complete partial fraction decomposition of f.
The decomposition
is returned as a (sorted) sequence Q consisting of triples, each of
which is of the form <d, k, n> where d is the denominator, k
is the multiplicity of the denominator, and n is the corresponding numerator,
and also d is irreducible and the degree of n is strictly less than
the degree of d.
Thus f equals the sum of the nt/(dt)>, where t ranges over the
triples contained in Q. If f is improper (the degree of its numerator
is greater than or equal to the degree of its denominator), then the first
triple of Q will be of the form <1, 1, q> where q is the quotient
of the numerator of f by the denominator of f.
Given a univariate rational function f in F=K(x), return the (unique)
complete squarefree partial fraction decomposition of f.
The decomposition
is returned as a (sorted) sequence Q consisting of triples, each of
which is of the form <d, k, n> where d is the denominator, k
is the multiplicity of the denominator, and n is the corresponding numerator,
and also d is squarefree and the degree of n is strictly less than
the degree of d.
Thus f equals the sum of the nt/(dt)>, where t ranges over the
triples contained in Q. If f is improper (the degree of its numerator
is greater than or equal to the degree of its denominator), then the first
triple of Q will be of the form <1, 1, q> where q is the quotient
of the numerator of f by the denominator of f.
We compute the squarefree and complete (irreducible) partial
fraction decompositions of a fraction in Q(t).
> F<t> := FunctionField(RationalField());
> P<x> := IntegerRing(F);
> f := ((t + 1)^8 - 1) / ((t^3 - 1)*(t + 1)^2*(t^2 - 4)^2);
> SD := SquarefreePartialFractionDecomposition(f);
> SD;
[
<x^4 + 2*x^3 - x - 2, 1, 467/196*x^3 + 1371/196*x^2 +
1391/196*x + 234/49>,
<x^2 - x - 2, 1, -271/196*x + 505/98>,
<x^2 - x - 2, 2, 271/14*x + 139/7>
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in SD] eq f;
> D := PartialFractionDecomposition(f);
> D;
[
<x - 2, 1, -3683/2646>,
<x - 2, 2, 410/63>,
<x - 1, 1, 85/36>,
<x + 1, 1, 1/108>,
<x + 1, 2, 1/18>,
<x + 2, 1, 1/18>,
<x^2 + x + 1, 1, -5/147*x - 8/147>
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in D] eq f;
true
Note that doing the same operation in the function field Z(t) must
modify the numerators and denominators to be integral but the result
is otherwise the same.
> F<t> := FunctionField(IntegerRing());
> P<x> := IntegerRing(F);
> f := ((t + 1)^8 - 1) / ((t^3 - 1)*(t + 1)^2*(t^2 - 4)^2);
> D := PartialFractionDecomposition(f);
> D;
[
<2646*x - 5292, 1, -3683>,
<63*x - 126, 2, 25830>,
<36*x - 36, 1, 85>,
<108*x + 108, 1, 1>,
<18*x + 18, 2, 18>,
<18*x + 36, 1, 1>,
<147*x^2 + 147*x + 147, 1, -5*x - 8>
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in D] eq f;
true
Finally, we compute the partial fraction decomposition of a fraction in
a function field whose coefficient ring is a multivariate function field.
> R<a, b> := FunctionField(IntegerRing(), 2);
> F<t> := FunctionField(R);
> P<x> := IntegerRing(F);
> f := 1 / ((t^2 - a)^2*(t + b)^2*t^3);
> SD := SquarefreePartialFractionDecomposition(f);
> SD;
[
<x^3 + b*x^2 - a*x - a*b, 1, (-3*a - 2*b^2)/(a^3*b^4)*x^2 +
(-a - 2*b^2)/(a^3*b^3)*x + (3*a + 3*b^2)/(a^2*b^4)>,
<x^3 + b*x^2 - a*x - a*b, 2, (a + b^2)/(a^2*b^3)*x^2 +
1/a^2*x + (-a - b^2)/(a*b^3)>,
<x, 1, (3*a + 2*b^2)/(a^3*b^4)>,
<x, 2, -2/(a^2*b^3)>,
<x, 3, 1/(a^2*b^2)>
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in SD] eq f;
true
> D := PartialFractionDecomposition(f);
> D;
[
<x, 1, (3*a + 2*b^2)/(a^3*b^4)>,
<x, 2, -2/(a^2*b^3)>,
<x, 3, 1/(a^2*b^2)>,
<x + b, 1, (-3*a + 7*b^2)/(a^3*b^4 - 3*a^2*b^6 + 3*a*b^8 -
b^10)>,
<x + b, 2, -1/(a^2*b^3 - 2*a*b^5 + b^7)>,
<x^2 - a, 1, (-3*a^2 - 3*a*b^2 + 2*b^4)/(a^6 - 3*a^5*b^2 +
3*a^4*b^4 - a^3*b^6)*x + (6*a*b - 2*b^3)/(a^5 - 3*a^4*b^2
+ 3*a^3*b^4 - a^2*b^6)>,
<x^2 - a, 2, (a + b^2)/(a^4 - 2*a^3*b^2 + a^2*b^4)*x -
2*b/(a^3 - 2*a^2*b^2 + a*b^4)>
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in D] eq f;
true
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