Difference polynomial rings

difference_polynomial_ring(R::Ring, elementary_symbols::Union{Vector{Symbol}, Int}, n_action_maps::Int) -> Tuple{DifferencePolyRing, Vector{DifferencePolyRingElem}}

Construct the difference polynomial ring over the base ring R with the given elementary symbols and n_action_maps commuting endomorphisms.

• If elementary_symbols is a vector of symbols, those names are used.
• If it is an integer m, the symbols u1, …, um are generated automatically.

In both cases, the jet variables that are initially available are those with jet [0,…,0], one for each elementary symbol.

This method returns a tuple (dpr, gens) where dpr is the resulting difference polynomial ring and gens is the vector of initial jet variables.

This constructor also accepts all keyword arguments of set_ranking! to control the ranking.

Examples

julia> R, variablesR = difference_polynomial_ring(QQ, 3, 4)
(Difference polynomial ring in 3 elementary symbols over QQ, DifferencePolyRingElem{QQFieldElem}[u1[0,0,0,0], u2[0,0,0,0], u3[0,0,0,0]])

julia> R
Difference polynomial ring in 3 elementary symbols u1, u2, u3
with 4 commuting endomorphisms
  over rational field

julia> variablesR
3-element Vector{DifferencePolyRingElem{QQFieldElem}}:
 u1[0,0,0,0]
 u2[0,0,0,0]
 u3[0,0,0,0]

julia> S, variablesS = difference_polynomial_ring(QQ, [:a, :b, :c], 4)
(Difference polynomial ring in 3 elementary symbols over QQ, DifferencePolyRingElem{QQFieldElem}[a[0,0,0,0], b[0,0,0,0], c[0,0,0,0]])

julia> S
Difference polynomial ring in 3 elementary symbols a, b, c
with 4 commuting endomorphisms
  over rational field

julia> variablesS
3-element Vector{DifferencePolyRingElem{QQFieldElem}}:
 a[0,0,0,0]
 b[0,0,0,0]
 c[0,0,0,0]

diff_action(p::DifferencePolyRingElem, i::Int)

Apply the i-th endomorphism to the polynomial p.

Examples

julia> dpr, (a,b,c) = difference_polynomial_ring(ZZ, [:a, :b, :c], 2); f = -2*a*b + 3*a*b^2;

julia> diff_action(3*a, 1)
3*a[1,0]

julia> diff_action(3*a, 2)
3*a[0,1]

julia> diff_action(f, 1)
(3*b[1,0]^2 - 2*b[1,0])*a[1,0]

julia> diff_action(f, 2)
(3*b[0,1]^2 - 2*b[0,1])*a[0,1]

diff_action(p::ActionPolyRingElem, d::Vector{Int}) -> ActionPolyRingElem

Successively apply the i-th diff-action d[i]-times to the polynomial p, where $i = 1, \ldots, \mathrm{length}(d)$.