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
```