#
** CycPols** —

*Module*.

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
```

#
** CycPols.subs** —

*Function*.

`subs(p::CycPol,v::Pol)`

a fast routine to compute `CycPol(p(v))`

but works for only two types of polynomials:

`v=Pol([e],1)`

for`e`

a`Root1`

, that is the value at`qe`

for`e=ζₙᵏ`

`v=Pol([1],n)`

that is the value at`qⁿ`

```
julia> p=CycPol(Pol()^2-1)
Φ₁Φ₂
julia> subs(p,Pol([E(3)],1))
ζ₃²Φ″₃Φ′₆
julia> subs(p,Pol()^2)
Φ₁Φ₂Φ₄
```

#
** CycPols.cyclotomic_polynomial** —

*Function*.

`cyclotomic_polynomial(n)`

returns the `n`

-th cyclotomic polynomial.

```
julia> cyclotomic_polynomial(5)
Pol{Int64}: q⁴+q³+q²+q+1
julia> cyclotomic_polynomial(24)
Pol{Int64}: q⁸-q⁴+1
```

#
** CycPols.CycPol** —

*Type*.

`CycPol`

s are internally a `struct`

with fields:

`.coeff`

: a coefficient, usually a `Cyc`

or a `Pol`

. The `Pol`

case allows to represent as `CycPol`

s arbitrary `Pol`

s which is useful sometimes.

`.valuation`

: an `Int`

.

`.v`

: a ModuleElt{Rational{Int},Int} where pairs `r=>m`

give the multiplicity `m`

of `Root1(;r=r)`

as a root.

So `CycPol(coeff,val,v)`

represents `coeff*q^val*prod((q-Root1(;r=r))^m for (r,m) in v)`

.