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 Pols.
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