CurrentModule = AbstractAlgebra
DocTestSetup = quote
using AbstractAlgebra
end
AbstractAlgebra.jl allows the creation of Puiseux series over any computable commutative
ring
Puiseux series are power series of the form
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.
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
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)
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 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,
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,
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 isequal(f, g), isequal(f, h) and
isequal(g, h) would all return false. However, if 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.
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
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.
coeff(a::Generic.PuiseuxSeriesElem, n::Int)
coeff(a::Generic.PuiseuxSeriesElem, n::Rational{Int})
Return the coefficient of the term of exponent
modulus{T <: ResElem}(::Generic.PuiseuxSeriesElem{T})
is_gen(::Generic.PuiseuxSeriesElem)
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 puiseux series field, 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)
Base.inv(::Generic.PuiseuxSeriesElem)
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(a::Generic.PuiseuxSeriesElem)
integral(a::Generic.PuiseuxSeriesElem)
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
Base.log(a::Generic.PuiseuxSeriesElem)
Base.exp(a::Generic.PuiseuxSeriesElem)
Base.sqrt(a::Generic.PuiseuxSeriesElem)
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 univariate polynomial ring, 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)