# Announcing new package `CycPols`

This package deals with products of Cyclotomic polynomials.

Cyclotomic numbers, and cyclotomic polynomials over the rationals or some
cyclotomic field, are important in the theories of finite reductive groups
and Spetses. In particular Schur elements of cyclotomic Hecke algebras are
products of cyclotomic polynomials.

The type `CycPol` represents the product of a `coeff` (a constant, a
polynomial or a rational fraction in one variable) with a rational fraction
in one variable with all poles or zeroes equal to 0 or roots of unity. The
advantages of representing as `CycPol` such objects are: nice display
(factorized), less storage, fast multiplication, division and evaluation.
The drawback is that addition and subtraction are not implemented!

This package uses the polynomials `Pol` defined by the package
`LaurentPolynomials` and the cyclotomic numbers `Cyc` defined by the
package `CyclotomicNumbers`.

The method `CycPol(a::Pol)` converts `a` to a `CycPol` by finding the
largest cyclotomic polynomial dividing, leaving a `Pol` `coefficient` if
some roots of the polynomial are not roots of unity.

``````julia> using LaurentPolynomials

julia> @Pol q
Pol{Int64}: q

julia> p=CycPol(q^25-q^24-2q^23-q^2+q+2) # a `Pol` coefficient remains
(q-2)Φ₁Φ₂Φ₂₃

julia> p(q) # evaluate CycPol p at q
Pol{Int64}: q²⁵-q²⁴-2q²³-q²+q+2

julia> p*inv(CycPol(q^2+q+1)) # `*`, `inv`, `/` and `//` are defined
(q-2)Φ₁Φ₂Φ₃⁻¹Φ₂₃

julia> -p  # one can multiply by a scalar
(-q+2)Φ₁Φ₂Φ₂₃

julia> valuation(p)
0

julia> degree(p)
25

julia> lcm(p,CycPol(q^3-1)) # lcm is fast between CycPols
(q-2)Φ₁Φ₂Φ₃Φ₂₃
``````
``````julia> print(p)
CycPol(Pol([-2, 1]),0,(1,0),(2,0),(23,0)) # a format which can be read in Julia
``````

Evaluating a `CycPol` at some `Pol` value gives in general a `Pol`. There
are exceptions where we can keep the value a `CycPol`: evaluating at
`Pol()^n` (that is `q^n`) or at `Pol([E(n,k)],1)` (that is `qζₙᵏ`). Then
`subs` gives that evaluation:

``````julia> subs(p,Pol()^-1) # evaluate as a CycPol at q⁻¹
(2-q⁻¹)q⁻²⁴Φ₁Φ₂Φ₂₃

julia> using CyclotomicNumbers

julia> subs(p,Pol([E(2)],1)) # or at -q
(-q-2)Φ₁Φ₂Φ₄₆
``````

The variable name used when printing a `CycPol` is the same as for `Pol`s.

When showing a `CycPol`, some factors over extension fields of the
cyclotomic polynomial `Φₙ` are given a special name. If `n` has a primitive
root `ξ`, `ϕ′ₙ` is the product of the `(q-ζ)` where `ζ` runs over the odd
powers of `ξ`, and `ϕ″ₙ` is the product for the even powers. Some further
factors are recognized for small `n`.

``````julia> CycPol(q^6-E(4))
Φ″₈Φ⁽¹³⁾₂₄
``````

The function `show_factors` gives the complete list of recognized factors
for a given `n`:

``````julia> CycPols.show_factors(24)
15-element Vector{Tuple{CycPol{Int64}, Pol}}:
(Φ₂₄, q⁸-q⁴+1)
(Φ′₂₄, q⁴+ζ₃²)
(Φ″₂₄, q⁴+ζ₃)
(Φ‴₂₄, q⁴-√2q³+q²-√2q+1)
(Φ⁗₂₄, q⁴+√2q³+q²+√2q+1)
(Φ⁽⁵⁾₂₄, q⁴-√6q³+3q²-√6q+1)
(Φ⁽⁶⁾₂₄, q⁴+√6q³+3q²+√6q+1)
(Φ⁽⁷⁾₂₄, q⁴+√-2q³-q²-√-2q+1)
(Φ⁽⁸⁾₂₄, q⁴-√-2q³-q²+√-2q+1)
(Φ⁽⁹⁾₂₄, q²+ζ₃²√-2q-ζ₃)
(Φ⁽¹⁰⁾₂₄, q²-ζ₃²√-2q-ζ₃)
(Φ⁽¹¹⁾₂₄, q²+ζ₃√-2q-ζ₃²)
(Φ⁽¹²⁾₂₄, q²-ζ₃√-2q-ζ₃²)
(Φ⁽¹³⁾₂₄, q⁴-ζ₄q²-1)
(Φ⁽¹⁴⁾₂₄, q⁴+ζ₄q²-1)
``````

Such a factor can be obtained directly as:

``````julia> CycPol(;conductor=24,no=7)
Φ⁽⁷⁾₂₄

julia> CycPol(;conductor=24,no=7)(q)
Pol{Cyc{Int64}}: q⁴+√-2q³-q²-√-2q+1
``````

This package also defines the function `cylotomic_polynomial`:

``````julia> p=cyclotomic_polynomial(24)
Pol{Int64}: q⁸-q⁴+1

julia> CycPol(p) # same as CycPol(;conductor=24,no=0)
Φ₂₄
``````
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