# Composition inverse of a power series

Avoid the symbolic ring whenever you can, and work in the appropriate structures for the task at hand.

## A worked out example

Manipulate polynomials and rational fractions in the corresponding structures. Define

sage: R.<z> = PolynomialRing(QQ)
sage: F = FractionField(R)
sage: S = PowerSeriesRing(QQ,'z')


and check the result:

sage: R
Univariate Polynomial Ring in z over Rational Field
sage: F
Fraction Field of Univariate Polynomial Ring in z over Rational Field
sage: S
Power Series Ring in z over Rational Field


Define your polynomials and rational fractions.

sage: a = 2 - 3*z + z^2
sage: b = z
sage: f = a / b


Check what you get.

sage: a
z^2 - 3*z + 2
sage: b
z
sage: f
(z^2 - 3*z + 2)/z


Check out where a, b, f live.

sage: a.parent()
Univariate Polynomial Ring in z over Rational Field
sage: b.parent()
Univariate Polynomial Ring in z over Rational Field
sage: f.parent()
Fraction Field of Univariate Polynomial Ring in z over Rational Field


Want the Taylor series at 1 of f, and its inverse under composition.

First shift the function to move to the point 1.

sage: g = f(1+z)
sage: g
(z^2 - z)/(z + 1)


Move to the ring of power series.

sage: gg = S(g)


Check the first few terms.

sage: gg + O(z^8)
-z + 2*z^2 - 2*z^3 + 2*z^4 - 2*z^5 + 2*z^6 - 2*z^7 + O(z^8)


The constant term is zero and the coefficient of z is nonzero: we can compute the composition inverse.

sage: hh = gg.reversion()


Examine the first few terms.

sage: hh + O(z^8)
-z + 2*z^2 - 6*z^3 + 22*z^4 - 90*z^5 + 394*z^6 - 1806*z^7 + O(z^8)


# Composition inverse of a power series

Avoid the symbolic ring whenever you can, and work in the appropriate structures for the task at hand.

## A worked out example

Manipulate polynomials and rational fractions in the corresponding structures. Define

sage: R.<z> = PolynomialRing(QQ)
sage: F = FractionField(R)
sage: S = PowerSeriesRing(QQ,'z')


and check the result:

sage: R
Univariate Polynomial Ring in z over Rational Field
sage: F
Fraction Field of Univariate Polynomial Ring in z over Rational Field
sage: S
Power Series Ring in z over Rational Field


Define your polynomials and rational fractions.

sage: a = 2 - 3*z + z^2
sage: b = z
sage: f = a / b


Check what you get.

sage: a
z^2 - 3*z + 2
sage: b
z
sage: f
(z^2 - 3*z + 2)/z


Check out where a, b, f live.

sage: a.parent()
Univariate Polynomial Ring in z over Rational Field
sage: b.parent()
Univariate Polynomial Ring in z over Rational Field
sage: f.parent()
Fraction Field of Univariate Polynomial Ring in z over Rational Field


Want the Taylor series at 1 of f, and its inverse under composition.

First shift the function to move to the point 1.

sage: g = f(1+z)
sage: g
(z^2 - z)/(z + 1)


Move to the ring of power series.

sage: gg = S(g)


Check the first few terms.

sage: gg + O(z^8)
-z + 2*z^2 - 2*z^3 + 2*z^4 - 2*z^5 + 2*z^6 - 2*z^7 + O(z^8)


The constant term is zero and the coefficient of z is nonzero: we can compute the composition inverse.

sage: hh = gg.reversion()


Examine the first few terms.

sage: hh + O(z^8)
-z + 2*z^2 - 6*z^3 + 22*z^4 - 90*z^5 + 394*z^6 - 1806*z^7 + O(z^8)


Check that hh is the composition inverse of gg:

sage: hh(gg)
z + O(z^21)
sage: gg(hh)
z + O(z^21)


# Composition inverse of a power series

Avoid the symbolic ring whenever you can, and work in the appropriate structures for the task at hand.

## A worked out example

Manipulate polynomials and rational fractions in the corresponding structures. Define

sage: R.<z> = PolynomialRing(QQ)
sage: F = FractionField(R)
sage: S = PowerSeriesRing(QQ,'z')


and check the result:

sage: R
Univariate Polynomial Ring in z over Rational Field
sage: F
Fraction Field of Univariate Polynomial Ring in z over Rational Field
sage: S
Power Series Ring in z over Rational Field


Define your polynomials and rational fractions.

sage: a = 2 - 3*z + z^2
sage: b = z
sage: f = a / b


Check what you get.

sage: a
z^2 - 3*z + 2
sage: b
z
sage: f
(z^2 - 3*z + 2)/z


Check out where a, b, f live.

sage: a.parent()
Univariate Polynomial Ring in z over Rational Field
sage: b.parent()
Univariate Polynomial Ring in z over Rational Field
sage: f.parent()
Fraction Field of Univariate Polynomial Ring in z over Rational Field


Want the Taylor series at 1 of f, and its inverse under composition.

First shift the function to move to the point 1.

sage: g = f(1+z)
sage: g
(z^2 - z)/(z + 1)


Move to the ring of power series.

sage: gg = S(g)


Check the first few terms.

sage: gg + O(z^8)
-z + 2*z^2 - 2*z^3 + 2*z^4 - 2*z^5 + 2*z^6 - 2*z^7 + O(z^8)


The constant term is zero and the coefficient of z is nonzero: we can compute the composition inverse.

sage: hh = gg.reversion()


Examine the first few terms.

sage: hh + O(z^8)
-z + 2*z^2 - 6*z^3 + 22*z^4 - 90*z^5 + 394*z^6 - 1806*z^7 + O(z^8)


Check that hh is the composition inverse of gg:

sage: hh(gg)
z + O(z^21)
sage: gg(hh)
z + O(z^21)


If you have a rational fraction g, then gnum = g.numerator() and gden = g.denominator() will get you the polynomials for the numerator and the denominator of g. You can check where they live by gnum.parent() and gden.parent(). If you need them to be in R, F or S, just do R(gnum), F(gnum), S(gnum), etc.