Add parametric ideal and interpolation-based algorithms

Author: wegank

src/AlgebraicSolving.jl

@@ -17,6 +17,11 @@ include("algorithms/decomposition.jl")
 include("algorithms/param-curve.jl")
 include("algorithms/hilbert.jl")
 include("algorithms/sampling.jl")
+#= param-ideal =#
+include("algorithms/param-ideal/groebner-bases.jl")
+include("algorithms/param-ideal/multiplication-matrices.jl")
+include("algorithms/param-ideal/hermite-matrices.jl")
+include("algorithms/param-ideal/parametrizations.jl")
 #= siggb =#
 include("siggb/siggb.jl")
 #= progress =#

src/algorithms/param-ideal/groebner-bases.jl

@@ -0,0 +1,229 @@
+function _map_exponent_vectors(f, p::MPolyRingElem, R::MPolyRing)::MPolyRingElem
+    cvzip = zip(coefficients(p), exponent_vectors(p))
+    M = MPolyBuildCtx(R)
+    for (c, v) in cvzip
+        push_term!(M, coefficient_ring(R)(c), f(v))
+    end
+    finish(M)
+end
+
+function _change_ring(p::MPolyRingElem, R::MPolyRing)::MPolyRingElem
+    n = number_of_variables(parent(p))
+    n′ = number_of_variables(R)
+    if n == 0 || n′ == 0
+        return R(constant_coefficient(p))
+    end
+    indices = Vector{Int}()
+    for s in symbols(R)
+        i = findfirst(==(s), symbols(parent(p)))
+        if isnothing(i)
+            push!(indices, 0)
+        else
+            push!(indices, i)
+        end
+    end
+    return _map_exponent_vectors(v -> [i == 0 ? 0 : v[i] for i in indices], p, R)
+end
+
+function _change_ring(v::Vector{<:MPolyRingElem}, R::MPolyRing)::Vector{<:MPolyRingElem}
+    map(p -> _change_ring(p, R), v)
+end
+
+function _change_ring(A::MatSpaceElem{<:MPolyRingElem}, R::MPolyRing)::MatSpaceElem{<:MPolyRingElem}
+    map(f -> _change_ring(f, R), A)
+end
+
+function _saturate(
+    f::Vector{QQMPolyRingElem},
+    g::QQMPolyRingElem;
+    worker_pool::AbstractWorkerPool=default_worker_pool()
+)::Vector{QQMPolyRingElem}
+    @assert !isempty(f)
+    R = parent(f[1])
+    n = number_of_variables(R)
+    R′, vars = polynomial_ring(QQ, [:sat; symbols(R)])
+    f′ = _change_ring(f, R′)
+    g′ = _change_ring(g, R′)
+    gb = AlgebraicSolving.groebner_basis(
+        AlgebraicSolving.Ideal([f′; vars[1] * g′ - 1]);
+        eliminate=1,
+        worker_pool=worker_pool
+    )
+    gb = _change_ring(gb, R)
+    gb
+end
+
+function _radical(
+    f::Vector{QQMPolyRingElem};
+    worker_pool::AbstractWorkerPool=default_worker_pool()
+)::Vector{QQMPolyRingElem}
+    @assert !isempty(f)
+    R = parent(f[1])
+    n = number_of_variables(R)
+    s = symbols(R)
+    rads = Vector{QQMPolyRingElem}()
+    for _ in 1:n
+        R′, _ = polynomial_ring(QQ, s, internal_ordering=internal_ordering(R))
+        disc = AlgebraicSolving.groebner_basis(
+            AlgebraicSolving.Ideal(_change_ring(f, R′));
+            eliminate=n - 1,
+            worker_pool=worker_pool
+        )
+        r = R′(1)
+        for (p, _) in factor(prod(disc))
+            r *= p
+        end
+        push!(rads, _change_ring(r, R))
+        s = [s[2:end]; s[1]]
+    end
+    gb = AlgebraicSolving.groebner_basis(
+        AlgebraicSolving.Ideal([f; rads]);
+        worker_pool=worker_pool
+    )
+    gb = _change_ring(gb, R)
+    gb
+end
+
+function groebner_basis(
+    I::ParametricIdeal{K},
+    vals::Vector{QQFieldElem};
+    worker_pool::AbstractWorkerPool=default_worker_pool()
+)::Vector{QQMPolyRingElem} where {K<:FracFieldElem}
+    gb_spec_hash = hash((I.gens, vals))
+    if haskey(I.gb_spec, gb_spec_hash)
+        return I.gb_spec[gb_spec_hash]
+    end
+    n = I.num_vars
+    elim = number_of_variables(parent(I.gens[1])) - n
+    R = parent(I.gens[1])
+    R₀, _ = polynomial_ring(QQ, symbols(R)[1:end], internal_ordering=:degrevlex)
+    R₁, _ = polynomial_ring(QQ, I.symbols, internal_ordering=:degrevlex)
+    fₓ = _change_ring(map(p -> map_coefficients(k -> evaluate(k, vals), p), I.gens), R₀)
+    gb = AlgebraicSolving.groebner_basis(
+        AlgebraicSolving.Ideal(fₓ);
+        eliminate=elim,
+        worker_pool=worker_pool
+    )
+    gb = _change_ring(gb, R₁)
+    for sat in I.sats
+        gb = _saturate(gb, _change_ring(map_coefficients(k -> evaluate(k, vals), sum(sat .^ 2)), R₁); worker_pool=worker_pool)
+    end
+    if I.radicalize
+        gb = _radical(gb; worker_pool=worker_pool)
+    end
+    gb = map(p -> p / leading_coefficient(p), gb)
+    lock(I.gb_spec_lock) do
+        I.gb_spec[gb_spec_hash] = gb
+    end
+    gb
+end
+
+function _is_zero_dimensional(gb::Vector{T})::Bool where {T<:MPolyRingElem}
+    if isempty(gb)
+        return false
+    end
+    R = parent(gb[1])
+    n = number_of_variables(R)
+    has_power = falses(n)
+    for f in gb
+        indices = var_indices(Nemo.leading_monomial(f))
+        if isempty(indices)
+            return true
+        end
+        if length(indices) == 1
+            has_power[indices[1]] = true
+        end
+    end
+    all(has_power)
+end
+
+function _monomial_basis(gb::Vector{T})::Vector{T} where {T<:MPolyRingElem}
+    if !_is_zero_dimensional(gb)
+        error("The ideal is not zero-dimensional, so a finite monomial basis does not exist.")
+    end
+    R = parent(gb[1])
+    n = number_of_variables(R)
+    lms = [Nemo.leading_monomial(f) for f in gb]
+    basis = typeof(gb[1])[]
+    queue = [one(R)]
+    while !isempty(queue)
+        u = popfirst!(queue)
+        if all(!divides(u, lm)[1] for lm in lms) && !(u in basis)
+            push!(basis, u)
+            for i in n:-1:1
+                push!(queue, u * gens(R)[i])
+            end
+        end
+    end
+    basis
+end
+
+function _expected_degree(
+    I::ParametricIdeal{K};
+    worker_pool::AbstractWorkerPool=default_worker_pool()
+)::Int where {K<:FracFieldElem}
+    gb = groebner_basis(I, QQ.(I.shift); worker_pool=worker_pool)
+    b = _monomial_basis(gb)
+    length(b)
+end
+
+@doc Markdown.doc"""
+    groebner_basis(I::ParametricIdeal{K}; <keyword arguments>) -> Vector{Generic.MPoly{FracFieldElem{QQMPolyRingElem}}}
+
+Computes a parametric Gröbner basis of the parametric ideal `I` w.r.t. the degree reverse lexicographic ordering. The Gröbner basis is computed by evaluating the ideal at a number of parameter values and interpolating the coefficients of the resulting Gröbner bases.
+
+# Arguments
+- `retry::Int=10`: the maximum number of consecutive failures allowed when computing the Gröbner basis or interpolating the coefficients.
+- `nr_thrds::Int=1`: the number of threads to use for parallel computations.
+- `worker_pool::AbstractWorkerPool=default_worker_pool()`: the worker pool to use for parallel computations.
+- `show_progress::Bool=false`: whether to show a progress bar while computing the Gröbner basis.
+"""
+function groebner_basis(
+    I::ParametricIdeal{K};
+    retry::Int=10,
+    nr_thrds::Int=1,
+    worker_pool::AbstractWorkerPool=default_worker_pool(),
+    show_progress::Bool=false
+)::Vector{Generic.MPoly{FracFieldElem{QQMPolyRingElem}}} where {K<:FracFieldElem}
+    input_hash = hash(I.gens)
+    if haskey(I.gb, input_hash)
+        return I.gb[input_hash]
+    end
+    R = parent(I.gens[1])
+    R₂ = base_ring(I.base_frac_field)
+    R₄, _ = polynomial_ring(I.base_frac_field, I.symbols, internal_ordering=internal_ordering(R))
+    gb_shift_mons = nothing
+    function bb(v)
+        gb = groebner_basis(I, v; worker_pool=worker_pool)
+        if !isnothing(gb_shift_mons)
+            gb_mons = [collect(monomials(p)) for p in gb]
+            if gb_mons != gb_shift_mons
+                error("Error evaluating black box function: the Gröbner basis has unexpected monomial structure at parameter values $(v).")
+            end
+        end
+        gb
+    end
+    gb_shift = bb(QQ.(I.shift))
+    gb_shift_mons = [collect(monomials(p)) for p in gb_shift]
+    pgb = fill(R₄(0), length(gb_shift))
+    len = length(gb_shift)
+    len_mons = length.(gb_shift)
+    for i in 1:len
+        for j in 1:len_mons[i]
+            denom = R₂(1)
+            c = Interpolation.cuyt_lee(
+                R₂, v -> evaluate(denom, v) * coeff(bb(v)[i], j);
+                initial_shift=I.shift,
+                retry=retry,
+                nr_thrds=nr_thrds,
+                show_progress=show_progress,
+                desc="Parametric Gröbner basis, size=($len,$(maximum(len_mons))), index=($i,$j)"
+            )
+            pgb[i] += R₄(c) / denom * _change_ring(Nemo.monomial(gb_shift[i], j), R₄)
+        end
+    end
+    lock(I.gb_lock) do
+        I.gb[input_hash] = pgb
+    end
+    pgb
+end