Generic Puiseux series
AbstractAlgebra.jl allows the creation of Puiseux series over any computable commutative ring $R$.
Puiseux series are power series of the form $a_jx^{j/m} + a_{j+1}x^{(j+1)/m} + \cdots + a_{k-1}x^{(k-1)/m} + O(x^{k/m})$ for some integer $m > 0$ where $i \geq 0$, $a_i \in R$ and the relative precision $k - j$ is at most equal to some specified precision $n$.
The generic Puiseux series module is implemented in src/generic/PuiseuxSeries.jl
.
As well as implementing the Series Ring interface, the Puiseux series module in AbstractAlgebra.jl implements the generic algorithms described below.
All of the generic functionality is part of the Generic
submodule of AbstractAlgebra.jl. This is exported by default so that it is not necessary to qualify function names.
Types and parent objects
The types of generic Puiseux series implemented by AbstractAlgebra.jl are Generic.PuiseuxSeriesRingElem{T}
and Generic.PuiseuxSeriesFieldElem{T}
.
Both series element types belong to the union type Generic.PuiseuxSeriesElem
.
Puiseux series elements belong directly to either RingElem
or FieldElem
since it is more useful to be able to distinguish whether they belong to a ring or field than it is to distinguish that they are Puiseux series.
The parent types for Puiseux series, Generic.PuiseuxSeriesRing{T}
and Generic.PuiseuxSeriesField{T}
respectively, belong to Ring
and Field
respectively.
The default precision, string representation of the variable and base ring $R$ of a generic Puiseux series are stored in its parent object.
Puiseux series ring constructors
In order to construct Puiseux series in AbstractAlgebra.jl, one must first construct the ring itself. This is accomplished with any of the following constructors.
puiseux_series_ring(R::Ring, prec_max::Int, s::VarName; cached::Bool = true)
puiseux_series_ring(R::Field, prec_max::Int, s::VarName; cached::Bool = true)
puiseux_series_field(R::Field, prec_max::Int, s::VarName; cached::Bool = true)
Given a base ring R
, a maximum relative precision and a string s
specifying how the generator (variable) should be printed, return a tuple S, x
representing the Puiseux series ring and its generator.
By default, S
will depend only on S
, x
and the maximum precision and will be cached. Setting the optional argument cached
to false
will prevent this.
Here are some examples of constructing various kinds of Puiseux series rings and coercing various elements into those rings.
Examples
julia> R, x = puiseux_series_ring(ZZ, 10, :x)
(Puiseux series ring in x over integers, x + O(x^11))
julia> S, y = puiseux_series_field(QQ, 10, :y)
(Puiseux series field in y over rationals, y + O(y^11))
julia> f = R()
O(x^10)
julia> g = S(123)
123 + O(y^10)
julia> h = R(BigInt(1234))
1234 + O(x^10)
julia> k = S(y + 1)
1 + y + O(y^10)
Big-oh notation
Series elements can be given a precision using the big-oh notation. This is provided by a function of the following form, (or something equivalent for Laurent series):
O(x::SeriesElem)
Examples
julia> R, x = puiseux_series_ring(ZZ, 10, :x)
(Puiseux series ring in x over integers, x + O(x^11))
julia> f = 1 + 2x + O(x^5)
1 + 2*x + O(x^5)
julia> g = 2x^(1//3) + 7x^(2//3) + O(x^(7//3))
2*x^(1//3) + 7*x^(2//3) + O(x^(7//3))
What is happening here in practice is that O(x^n)
is creating the series 0 + O(x^n)
and the rules for addition of series dictate that if this is added to a series of greater precision, then the lower of the two precisions must be used.
Of course it may be that the precision of the series that O(x^n)
is added to is already lower than n
, in which case adding O(x^n)
has no effect. This is the case if the default precision is too low, since x
on its own has the default precision.
Puiseux series implementation
Puiseux series have their maximum relative precision capped at some value prec_max
. This refers to the internal Laurent series used to store the Puiseux series, i.e. the series without denominators in the exponents.
The Puiseux series type stores such a Laurent series and a scale
or denominator for the exponents. For example, $f(x) = 1 + x^{1/3} + 2x^{2/3} + O(x^{7/3})$ would be stored as a Laurent series $1 + x + 2x^2 + O(x^7)$ and a scale of $3$..
The maximum precision is also used as a default (Laurent) precision in the case of coercing coefficients into the ring and for any computation where the result could mathematically be given to infinite precision.
In all models we say that two Puiseux series are equal if they agree up to the minimum absolute precision of the two power series.
Thus, for example, $x^5 + O(x^{10}) == 0 + O(x^5)$, since the minimum absolute precision is $5$.
Sometimes it is necessary to compare Puiseux series not just for arithmetic equality, as above, but to see if they have precisely the same precision and terms. For this purpose we introduce the isequal
function.
For example, if $f = x^2 + O(x^7)$ and $g = x^2 + O(x^8)$ and $h = 0 + O(x^2)$ then $f == g$, $f == h$ and $g == h$, but isequal(f, g)
, isequal(f, h)
and isequal(g, h)
would all return false
. However, if $k = x^2 + O(x^7)$ then isequal(f, k)
would return true
.
There are a number of technicalities that must be observed when working with Puiseux series. As these are the same as for the other series rings in AbstractAlgebra.jl, we refer the reader to the documentation of series rings for information about these issues.
Basic ring functionality
All Puiseux series provide the functionality described in the Ring and Series Ring interfaces with the exception of the pol_length
and polcoeff
functions. Naturally the set_precision!
, set_valuation!
and coeff
functions can take a rational exponent.
Examples
julia> S, x = puiseux_series_ring(ZZ, 10, :x)
(Puiseux series ring in x over integers, x + O(x^11))
julia> f = 1 + 3x + x^3 + O(x^10)
1 + 3*x + x^3 + O(x^10)
julia> g = 1 + 2x^(1//3) + x^(2//3) + O(x^(7//3))
1 + 2*x^(1//3) + x^(2//3) + O(x^(7//3))
julia> h = zero(S)
O(x^10)
julia> k = one(S)
1 + O(x^10)
julia> isone(k)
true
julia> iszero(f)
false
julia> coeff(g, 1//3)
2
julia> U = base_ring(S)
Integers
julia> v = var(S)
:x
julia> T = parent(x + 1)
Puiseux series ring in x over integers
julia> g == deepcopy(g)
true
julia> t = divexact(2g, 2)
1 + 2*x^(1//3) + x^(2//3) + O(x^(7//3))
julia> p = precision(f)
10//1
Puiseux series functionality provided by AbstractAlgebra.jl
The functionality below is automatically provided by AbstractAlgebra.jl for any Puiseux series.
Of course, modules are encouraged to provide specific implementations of the functions described here, that override the generic implementation.
Basic functionality
coeff(a::Generic.PuiseuxSeriesElem, n::Int)
coeff(a::Generic.PuiseuxSeriesElem, n::Rational{Int})
Return the coefficient of the term of exponent $n$ of the given power series. If $n$ exceeds the current precision of the power series or does not correspond to a nonzero term of the Puiseux series, the function returns a zero coefficient.
AbstractAlgebra.modulus
— Methodmodulus(a::Generic.PuiseuxSeriesElem{T}) where {T <: ResElem}
Return the modulus of the coefficients of the given Puiseux series.
AbstractAlgebra.is_gen
— Methodis_gen(a::Generic.PuiseuxSeriesElem)
Return true
if the given Puiseux series is arithmetically equal to the generator of its Puiseux series ring to its current precision, otherwise return false
.
Examples
julia> R, t = puiseux_series_ring(QQ, 10, :t)
(Puiseux series field in t over rationals, t + O(t^11))
julia> S, x = puiseux_series_ring(R, 30, :x)
(Puiseux series field in x over R, x + O(x^31))
julia> a = O(x^4)
O(x^4)
julia> b = (t + 3)*x + (t^2 + 1)*x^2 + O(x^4)
(3 + t + O(t^10))*x + (1 + t^2 + O(t^10))*x^2 + O(x^4)
julia> k = is_gen(gen(R))
true
julia> m = is_unit(-1 + x^(1//3) + 2x^2)
true
julia> n = valuation(a)
4//1
julia> p = valuation(b)
1//1
julia> c = coeff(b, 2)
1 + t^2 + O(t^10)
Division
Base.inv
— Methodinv(M::MatrixElem{T}) where {T <: RingElement}
Given a non-singular $n\times n$ matrix over a ring, return an $n\times n$ matrix $X$ such that $MX = I_n$, where $I_n$ is the $n\times n$ identity matrix. If $M$ is not invertible over the base ring an exception is raised.
Base.inv(a::PuiseuxSeriesElem{T}) where T <: RingElement
Return the inverse of the power series $a$, i.e. $1/a$, if it exists. Otherwise an exception is raised.
inv(a::LocalizedEuclideanRingElem{T}, checked::Bool = true) where {T <: RingElem}
Returns the inverse element of $a$ if $a$ is a unit. If 'checked = false' the invertibility of $a$ is not checked and the corresponding inverse element of the Fraction Field is returned.
Examples
julia> R, x = puiseux_series_ring(QQ, 30, :x)
(Puiseux series field in x over rationals, x + O(x^31))
julia> a = 1 + x + 2x^2 + O(x^5)
1 + x + 2*x^2 + O(x^5)
julia> b = R(-1)
-1 + O(x^30)
julia> c = inv(a)
1 - x - x^2 + 3*x^3 - x^4 + O(x^5)
julia> d = inv(b)
-1 + O(x^30)
Derivative and integral
AbstractAlgebra.derivative
— Methodderivative(f::AbsPowerSeriesRingElem{T})
Return the derivative of the power series $f$.
derivative(f::RelPowerSeriesRingElem{T})
Return the derivative of the power series $f$.
julia> R, x = power_series_ring(QQ, 10, :x)
(Univariate power series ring in x over Rationals, x + O(x^11))
julia> f = 2 + x + 3x^3
2 + x + 3*x^3 + O(x^10)
julia> derivative(f)
1 + 9*x^2 + O(x^9)
derivative(f::MPolyRingElem{T}, j::Int) where {T <: RingElement}
Return the partial derivative of f
with respect to $j$-th variable of the polynomial ring.
derivative(f::MPolyRingElem{T}, x::MPolyRingElem{T}) where T <: RingElement
Return the partial derivative of f
with respect to x
. The value x
must be a generator of the polynomial ring of f
.
derivative(a::Generic.PuiseuxSeriesElem{T}) where T <: RingElement
Return the derivative of the given Puiseux series $a$.
AbstractAlgebra.integral
— Methodintegral(f::AbsPowerSeriesRingElem{T})
Return the integral of the power series $f$.
integral(f::RelPowerSeriesRingElem{T})
Return the integral of the power series $f$.
julia> R, x = power_series_ring(QQ, 10, :x)
(Univariate power series ring in x over Rationals, x + O(x^11))
julia> f = 2 + x + 3x^3
2 + x + 3*x^3 + O(x^10)
julia> integral(f)
2*x + 1//2*x^2 + 3//4*x^4 + O(x^11)
integral(a::Generic.PuiseuxSeriesElem{T}) where T <: RingElement
Return the integral of the given Puiseux series $a$.
Examples
julia> R, x = puiseux_series_ring(QQ, 10, :x)
(Puiseux series field in x over rationals, x + O(x^11))
julia> f = x^(5//3) + x^(7//3) + x^(11//3)
x^(5//3) + x^(7//3) + x^(11//3) + O(x^5)
julia> derivative(f)
5//3*x^(2//3) + 7//3*x^(4//3) + 11//3*x^(8//3) + O(x^4)
julia> derivative(integral(f)) == f
true
Special functions
Base.log
— Methodlog(a::Generic.PuiseuxSeriesElem{T}) where T <: RingElement
Return the logarithm of the given Puiseux series $a$.
Base.exp
— Methodexp(a::AbsPowerSeriesRingElem)
Return the exponential of the power series $a$.
exp(a::RelPowerSeriesRingElem)
Return the exponential of the power series $a$.
exp(a::Generic.LaurentSeriesElem)
Return the exponential of the power series $a$.
exp(a::Generic.PuiseuxSeriesElem{T}) where T <: RingElement
Return the exponential of the given Puiseux series $a$.
Base.sqrt
— MethodBase.sqrt(f::PolyRingElem{T}; check::Bool=true) where T <: RingElement
Return the square root of $f$. By default the function checks the input is square and raises an exception if not. If check=false
this check is omitted.
Base.sqrt(a::FracElem{T}; check::Bool=true) where T <: RingElem
Return the square root of $a$. By default the function will throw an exception if the input is not square. If check=false
this test is omitted.
sqrt(a::Generic.PuiseuxSeriesElem{T}; check::Bool=true) where T <: RingElement
Return the square root of the given Puiseux series $a$. By default the function will throw an exception if the input is not square. If check=false
this test is omitted.
Examples
julia> R, t = polynomial_ring(QQ, :t)
(Univariate polynomial ring in t over rationals, t)
julia> S, x = puiseux_series_ring(R, 30, :x)
(Puiseux series ring in x over R, x + O(x^31))
julia> T, z = puiseux_series_ring(QQ, 30, :z)
(Puiseux series field in z over rationals, z + O(z^31))
julia> a = 1 + z + 3z^2 + O(z^5)
1 + z + 3*z^2 + O(z^5)
julia> b = z + 2z^2 + 5z^3 + O(z^5)
z + 2*z^2 + 5*z^3 + O(z^5)
julia> c = exp(x + O(x^40))
1 + x + 1//2*x^2 + 1//6*x^3 + 1//24*x^4 + 1//120*x^5 + 1//720*x^6 + 1//5040*x^7 + 1//40320*x^8 + 1//362880*x^9 + 1//3628800*x^10 + 1//39916800*x^11 + 1//479001600*x^12 + 1//6227020800*x^13 + 1//87178291200*x^14 + 1//1307674368000*x^15 + 1//20922789888000*x^16 + 1//355687428096000*x^17 + 1//6402373705728000*x^18 + 1//121645100408832000*x^19 + 1//2432902008176640000*x^20 + 1//51090942171709440000*x^21 + 1//1124000727777607680000*x^22 + 1//25852016738884976640000*x^23 + 1//620448401733239439360000*x^24 + 1//15511210043330985984000000*x^25 + 1//403291461126605635584000000*x^26 + 1//10888869450418352160768000000*x^27 + 1//304888344611713860501504000000*x^28 + 1//8841761993739701954543616000000*x^29 + 1//265252859812191058636308480000000*x^30 + O(x^31)
julia> d = divexact(x, exp(x + O(x^40)) - 1)
1 - 1//2*x + 1//12*x^2 - 1//720*x^4 + 1//30240*x^6 - 1//1209600*x^8 + 1//47900160*x^10 - 691//1307674368000*x^12 + 1//74724249600*x^14 - 3617//10670622842880000*x^16 + 43867//5109094217170944000*x^18 - 174611//802857662698291200000*x^20 + 77683//14101100039391805440000*x^22 - 236364091//1693824136731743669452800000*x^24 + 657931//186134520519971831808000000*x^26 - 3392780147//37893265687455865519472640000000*x^28 + O(x^29)
julia> f = exp(b)
1 + z + 5//2*z^2 + 43//6*z^3 + 193//24*z^4 + O(z^5)
julia> h = sqrt(a)
1 + 1//2*z + 11//8*z^2 - 11//16*z^3 - 77//128*z^4 + O(z^5)