Puiseux Series Ring Element

A Puiseux series is a series of the form

\[p(x) = \sum_{n=N}^{\infty} a_n (x-a)^{n/e},\]

where the integer \(e\) is called the ramification index of the series and the number \(a\) is the center. A Puiseux series is essentially a Laurent series but with fractional exponents.

EXAMPLES:

We begin by constructing the ring of Puiseux series in \(x\) with coefficients in the rationals:

sage: R.<x> = PuiseuxSeriesRing(QQ)

This command also defines x as the generator of this ring.

When constructing a Puiseux series, the ramification index is automatically determined from the greatest common divisor of the exponents:

sage: p = x^(1/2); p
x^(1/2)
sage: p.ramification_index()
2
sage: q = x^(1/2) + x**(1/3); q
x^(1/3) + x^(1/2)
sage: q.ramification_index()
6

Other arithmetic can be performed with Puiseux Series:

sage: p + q
x^(1/3) + 2*x^(1/2)
sage: p - q
-x^(1/3)
sage: p * q
x^(5/6) + x
sage: (p / q).add_bigoh(4/3)
x^(1/6) - x^(1/3) + x^(1/2) - x^(2/3) + x^(5/6) - x + x^(7/6) + O(x^(4/3))

Mind the base ring. However, the base ring can be changed:

sage: I*q
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Number Field in I with defining polynomial x^2 + 1 with I = 1*I' and 'Puiseux Series Ring in x over Rational Field'
sage: qz = q.change_ring(ZZ); qz
x^(1/3) + x^(1/2)
sage: qz.parent()
Puiseux Series Ring in x over Integer Ring

Other properties of the Puiseux series can be easily obtained:

sage: r = (3*x^(-1/5) + 7*x^(2/5) + (1/2)*x).add_bigoh(6/5); r
3*x^(-1/5) + 7*x^(2/5) + 1/2*x + O(x^(6/5))
sage: r.valuation()
-1/5
sage: r.prec()
6/5
sage: r.precision_absolute()
6/5
sage: r.precision_relative()
7/5
sage: r.exponents()
[-1/5, 2/5, 1]
sage: r.coefficients()
[3, 7, 1/2]

Finally, Puiseux series are compatible with other objects in Sage. For example, you can perform arithmetic with Laurent series:

sage: L.<x> = LaurentSeriesRing(ZZ)
sage: l = 3*x^(-2) + x^(-1) + 2 + x**3
sage: r + l
3*x^-2 + x^-1 + 3*x^(-1/5) + 2 + 7*x^(2/5) + 1/2*x + O(x^(6/5))

AUTHORS:

REFERENCES:

class sage.rings.puiseux_series_ring_element.PuiseuxSeries

Bases: sage.structure.element.AlgebraElement

A Puiseux series.

\[\sum_{n=-N}^\infty a_n x^{n/e}\]

It is stored as a Laurent series:

\[\sum_{n=-N}^\infty a_n t^n\]

where \(t = x^{1/e}\).

INPUT:

  • parent – the parent ring

  • f – one of the following types of inputs:

    • instance of PuiseuxSeries

    • instance that can be coerced into the Laurent series ring of the parent

  • e – integer (default: 1) the ramification index

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(1/2) + x**3; p
x^(1/2) + x^3
sage: q = x**(1/2) - x**(-1/2)
sage: r = q.add_bigoh(7/2); r
-x^(-1/2) + x^(1/2) + O(x^(7/2))
sage: r**2
x^-1 - 2 + x + O(x^3)
add_bigoh(prec)

Return the truncated series at chosen precision prec.

INPUT:

  • prec – the precision of the series as a rational number

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(-7/2) + 3 + 5*x^(1/2) - 7*x**3
sage: p.add_bigoh(2)
x^(-7/2) + 3 + 5*x^(1/2) + O(x^2)
sage: p.add_bigoh(0)
x^(-7/2) + O(1)
sage: p.add_bigoh(-1)
x^(-7/2) + O(x^-1)

Note

The precision passed to the method is adapted to the common ramification index:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = x**(-1/3) + 2*x**(1/5)
sage: p.add_bigoh(1/2)
x^(-1/3) + 2*x^(1/5) + O(x^(7/15))
change_ring(R)

Return self over a the new ring R.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = x^(-7/2) + 3 + 5*x^(1/2) - 7*x**3
sage: q = p.change_ring(QQ); q
x^(-7/2) + 3 + 5*x^(1/2) - 7*x^3
sage: q.parent()
Puiseux Series Ring in x over Rational Field
coefficients()

Return the list of coefficients.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = x^(3/4) + 2*x^(4/5) + 3* x^(5/6)
sage: p.coefficients()
[1, 2, 3]
common_prec(p)

Return the minimum precision of \(p\) and self.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = (x**(-1/3) + 2*x**3)**2
sage: q5 = p.add_bigoh(5); q5
x^(-2/3) + 4*x^(8/3) + O(x^5)
sage: q7 = p.add_bigoh(7); q7
x^(-2/3) + 4*x^(8/3) + 4*x^6 + O(x^7)
sage: q5.common_prec(q7)
5
sage: q7.common_prec(q5)
5
degree()

Return the degree of self.

EXAMPLES:

sage: P.<y> = PolynomialRing(GF(5))
sage: R.<x> = PuiseuxSeriesRing(P)
sage: p = 3*y*x**(-2/3) + 2*y**2*x**(1/5); p
3*y*x^(-2/3) + 2*y^2*x^(1/5)
sage: p.degree()
1/5
exponents()

Return the list of exponents.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = x^(3/4) + 2*x^(4/5) + 3* x^(5/6)
sage: p.exponents()
[3/4, 4/5, 5/6]
inverse()

Return the inverse of self.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(-7/2) + 3 + 5*x^(1/2) - 7*x**3
sage: 1/p
x^(7/2) - 3*x^7 - 5*x^(15/2) + 7*x^10 + 9*x^(21/2) + 30*x^11 +
25*x^(23/2) + O(x^(27/2))
is_monomial()

Return whether self is a monomial.

This is True if and only if self is \(x^p\) for some rational \(p\).

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(1/2) + 3/4 * x^(2/3)
sage: p.is_monomial()
False
sage: q = x**(11/13)
sage: q.is_monomial()
True
sage: q = 4*x**(11/13)
sage: q.is_monomial()
False
is_unit()

Return whether self is a unit.

A Puiseux series is a unit if and only if its leading coefficient is.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = x^(-7/2) + 3 + 5*x^(1/2) - 7*x**3
sage: p.is_unit()
True
sage: q = 4 * x^(-7/2) + 3 * x**4
sage: q.is_unit()
False
is_zero()

Return whether self is zero.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(1/2) + 3/4 * x^(2/3)
sage: p.is_zero()
False
sage: R.zero().is_zero()
True
laurent_part()

Return the underlying Laurent series.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(1/2) + 3/4 * x^(2/3)
sage: p.laurent_part()
x^3 + 3/4*x^4
laurent_series()

If self is a Laurent series, return it as a Laurent series.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = x**(1/2) - x**(-1/2)
sage: p.laurent_series()
Traceback (most recent call last):
...
ArithmeticError: self is not a Laurent series
sage: q = p**2
sage: q.laurent_series()
x^-1 - 2 + x
list()

Return the list of coefficients indexed by the exponents of the the corresponding Laurent series.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = x^(3/4) + 2*x^(4/5) + 3* x^(5/6)
sage: p.list()
[1, 0, 0, 2, 0, 3]
power_series()

If self is a power series, return it as a power series.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQbar)
sage: p = x**(3/2) - QQbar(I)*x**(1/2)
sage: p.power_series()
Traceback (most recent call last):
...
ArithmeticError: self is not a power series
sage: q = p**2
sage: q.power_series()
-x - 2*I*x^2 + x^3
prec()

Return the precision of self.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = (x**(-1/3) + 2*x**3)**2; p
x^(-2/3) + 4*x^(8/3) + 4*x^6
sage: q = p.add_bigoh(5); q
x^(-2/3) + 4*x^(8/3) + O(x^5)
sage: q.prec()
5
precision_absolute()

Return the precision of self.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = (x**(-1/3) + 2*x**3)**2; p
x^(-2/3) + 4*x^(8/3) + 4*x^6
sage: q = p.add_bigoh(5); q
x^(-2/3) + 4*x^(8/3) + O(x^5)
sage: q.prec()
5
precision_relative()

Return the relative precision of the series.

The relative precision of the Puiseux series is the difference between its absolute precision and its valuation.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(GF(3))
sage: p = (x**(-1/3) + 2*x**3)**2; p
x^(-2/3) + x^(8/3) + x^6
sage: q = p.add_bigoh(7); q
x^(-2/3) + x^(8/3) + x^6 + O(x^7)
sage: q.precision_relative()
23/3
ramification_index()

Return the ramification index.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(1/2) + 3/4 * x^(2/3)
sage: p.ramification_index()
6
shift(r)

Return this Puiseux series multiplied by \(x^r\).

EXAMPLES:

sage: P.<y> = LaurentPolynomialRing(ZZ)
sage: R.<x> = PuiseuxSeriesRing(P)
sage: p = y*x**(-1/3) + 2*y^(-2)*x**(1/2); p
y*x^(-1/3) + (2*y^-2)*x^(1/2)
sage: p.shift(3)
y*x^(8/3) + (2*y^-2)*x^(7/2)
truncate(r)

Return the Puiseux series of degree \(< r\).

This is equivalent to self modulo \(x^r\).

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(ZZ)
sage: p = (x**(-1/3) + 2*x**3)**2; p
x^(-2/3) + 4*x^(8/3) + 4*x^6
sage: q = p.truncate(5); q
x^(-2/3) + 4*x^(8/3)
sage: q == p.add_bigoh(5)
True
valuation()

Return the valuation of self.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(-7/2) + 3 + 5*x^(1/2) - 7*x**3
sage: p.valuation()
-7/2
variable()

Return the variable of self.

EXAMPLES:

sage: R.<x> = PuiseuxSeriesRing(QQ)
sage: p = x^(-7/2) + 3 + 5*x^(1/2) - 7*x**3
sage: p.variable()
'x'