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Find dimension of solution variety

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This commit is contained in:
Aaron Fenyes 2024-01-30 02:45:14 -05:00
parent 6349f298ae
commit 4e02ee16fc
2 changed files with 25 additions and 21 deletions
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31
engine-proto/engine.jl
View File

@ -1,3 +1,5 @@
include("hitting-set.jl")
module Engine
export Construction, mprod
@ -6,27 +8,25 @@ import Subscripts
using LinearAlgebra
using AbstractAlgebra
using Groebner
using ..HittingSet
# --- commutative algebra ---
# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate
# polynomial rings when coefficients are integers. in `reduce_gens`, the
# `lmnode` constructor requires < to be defined on the coefficients, and the
# `reducer_size` heuristic requires `ndigits` to be defined on the coefficients.
# this patch for `reducer_size` removes the `ndigits` dependency
##function Generic.reducer_size(f::T) where {U <: MPolyRingElem{<:FieldElement}, V, N, T <: Generic.lmnode{U, V, N}}
## if f.size != 0.0
## return f.size
## end
## return 0.0 + sum(j^2 for j in 1:length(f.poly))
##end
# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate
# polynomial rings when the coefficients are integers. we use Groebner to extend
# support to rationals and to finite fields of prime order
Generic.reduce_gens(I::Generic.Ideal{U}) where {T <: FieldElement, U <: MPolyRingElem{T}} =
Generic.Ideal{U}(base_ring(I), groebner(gens(I)))
function codimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}}
leading = [exponent_vector(f, 1) for f in gens(I)]
targets = [Set(findall(.!iszero.(exp_vec))) for exp_vec in leading]
length(HittingSet.solve(HittingSetProblem(targets), maxdepth))
end
dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} =
length(gens(base_ring(I))) - codimension(I, maxdepth)
# --- primitve elements ---
abstract type Element{T} end
@ -183,6 +183,7 @@ s = Engine.Sphere{CoeffType}()
a_on_s = Engine.LiesOn{CoeffType}(a, s)
ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
ideal_a_s = Engine.realize(ctx)
println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freeom")
b = Engine.Point{CoeffType}()
b_on_s = Engine.LiesOn{CoeffType}(b, s)
@ -190,6 +191,7 @@ Engine.push!(ctx, b)
Engine.push!(ctx, s)
Engine.push!(ctx, b_on_s)
ideal_ab_s = Engine.realize(ctx)
println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freeom")
spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
tangencies = [
@ -200,5 +202,6 @@ tangencies = [
)
for n in 1:3
]
ctx_chain = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
ideal_chain = Engine.realize(ctx_chain)
ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
ideal_tan_sph = Engine.realize(ctx_tan_sph)
println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freeom")

15
engine-proto/hitting-set.jl
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@ -1,13 +1,15 @@
module HittingSet
export HittingSetProblem, solve
HittingSetProblem{T} = Pair{Set{T}, Vector{Pair{T, Set{Set{T}}}}}
# `subsets` should be a collection of Set objects
function HittingSetProblem(subsets, chosen = Set())
wholeset = union(subsets...)
# `targets` should be a collection of Set objects
function HittingSetProblem(targets, chosen = Set())
wholeset = union(targets...)
T = eltype(wholeset)
unsorted_moves = [
elt => Set(filter(s -> elt ∉ s, subsets))
elt => Set(filter(s -> elt ∉ s, targets))
for elt in wholeset
]
moves = sort(unsorted_moves, by = pair -> length(pair.second))
@ -32,7 +34,6 @@ end
function solve(pblm::HittingSetProblem{T}, maxdepth = Inf) where T
problems = Dict(pblm)
println(typeof(problems))
while length(first(problems).first) < maxdepth
subproblems = typeof(problems)()
for (chosen, moves) in problems
@ -56,7 +57,7 @@ end
function test(n = 1)
T = [Int64, Int64, Symbol, Symbol][n]
subsets = Set{T}.([
targets = Set{T}.([
[
[1, 3, 5],
[2, 3, 4],
@ -98,7 +99,7 @@ function test(n = 1)
[:b, :z, :t14]
]
][n])
problem = HittingSetProblem(subsets)
problem = HittingSetProblem(targets)
if isa(problem, HittingSetProblem{T})
println("Correct type")
else