From 93e681441590d5c77da99f0ba754642909f7bf9e Mon Sep 17 00:00:00 2001
From: Weijia Wang <9713184+wegank@users.noreply.github.com>
Date: Fri, 20 Mar 2026 15:50:12 +0100
Subject: [PATCH] Add CAD-based sample points

---
 src/AlgebraicSolving.jl     |   1 +
 src/algorithms/sampling.jl  | 228 ++++++++++++++++++++++++++++++++++++
 src/exports.jl              |   1 +
 src/imports.jl              |   2 +-
 test/algorithms/sampling.jl |  37 ++++++
 test/runtests.jl            |   1 +
 6 files changed, 269 insertions(+), 1 deletion(-)
 create mode 100644 src/algorithms/sampling.jl
 create mode 100644 test/algorithms/sampling.jl

diff --git a/src/AlgebraicSolving.jl b/src/AlgebraicSolving.jl
index f48e3d3..846f376 100644
--- a/src/AlgebraicSolving.jl
+++ b/src/AlgebraicSolving.jl
@@ -16,6 +16,7 @@ include("algorithms/dimension.jl")
 include("algorithms/decomposition.jl")
 include("algorithms/param-curve.jl")
 include("algorithms/hilbert.jl")
+include("algorithms/sampling.jl")
 #= siggb =#
 include("siggb/siggb.jl")
 #= progress =#
diff --git a/src/algorithms/sampling.jl b/src/algorithms/sampling.jl
new file mode 100644
index 0000000..a3110e4
--- /dev/null
+++ b/src/algorithms/sampling.jl
@@ -0,0 +1,228 @@
+@doc Markdown.doc"""
+    _to_multivariate(R::QQMPolyRing, p::QQPolyRingElem)::QQMPolyRingElem
+
+Convert a univariate polynomial `p` to a multivariate polynomial in the ring `R` with the same variable.
+
+**Note**: This is an internal function.
+"""
+function _to_multivariate(R::QQMPolyRing, p::QQPolyRingElem)::QQMPolyRingElem
+    @assert length(gens(R)) == 1
+    M = MPolyBuildCtx(R)
+    for i in 0:length(p)
+        push_term!(M, coeff(p, i), [i])
+    end
+    finish(M)
+end
+
+@doc Markdown.doc"""
+    _sample_points(f::Vector{QQPolyRingElem}; worker_pool::AbstractWorkerPool=default_worker_pool())::Vector{QQFieldElem}
+
+Given a vector of univariate polynomials, returns a vector of rational numbers such that each connected component of the non-vanishing set of the polynomials contains at least one of the returned points.
+
+**Note**: This is an internal function.
+"""
+function _sample_points(f::Vector{QQPolyRingElem}; worker_pool::AbstractWorkerPool=default_worker_pool())::Vector{QQFieldElem}
+    R, _ = polynomial_ring(QQ, [:x])
+    fₓ = map(p -> _to_multivariate(R, p), f)
+    factors = unique([p for fₓ′ in fₓ for (p, _) in factor(fₓ′)])
+    # We map each factor to its roots, and each root to its factor
+    roots_by_factor = Dict{QQMPolyRingElem,Vector{Vector{Vector{QQFieldElem}}}}()
+    factors_by_root = Dict{Vector{Vector{QQFieldElem}},QQMPolyRingElem}()
+    for factor in factors
+        roots = real_solutions(Ideal([factor]); interval=true, worker_pool=worker_pool)
+        roots_by_factor[factor] = roots
+        for r in roots
+            factors_by_root[r] = factor
+        end
+    end
+    # We order intervals by their left endpoint
+    roots = sort(collect(keys(factors_by_root)), by=x -> x[1][1])
+    # If intervals are not ordered by their right endpoint,
+    # we merge the offending factors
+    while (!issorted(roots, by=x -> x[1][2]))
+        i = findfirst(i -> roots[i][1][2] > roots[i+1][1][2], 1:length(roots)-1)
+        bad_factors = [factors_by_root[roots[i]], factors_by_root[roots[i+1]]]
+        # Remove old factors and roots
+        for bad_factor in bad_factors
+            for r in roots_by_factor[bad_factor]
+                delete!(factors_by_root, r)
+            end
+            delete!(roots_by_factor, bad_factor)
+        end
+        # Replace with merged factor and its roots
+        merged_factor = prod(bad_factors)
+        merged_roots = real_solutions(Ideal([merged_factor]); interval=true, worker_pool=worker_pool)
+        roots_by_factor[merged_factor] = merged_roots
+        for r in merged_roots
+            factors_by_root[r] = merged_factor
+        end
+        roots = sort(collect(keys(factors_by_root)), by=r -> r[1][1])
+    end
+    # Now the intervals are properly ordered, we can sample points
+    points = Vector{QQFieldElem}()
+    if length(roots) == 0
+        push!(points, QQ(0))
+        return points
+    end
+    push!(points, floor(roots[1][1][1]) - 1)
+    for i in 1:length(roots)-1
+        push!(points, (roots[i][1][2] + roots[i+1][1][1]) // 2)
+    end
+    push!(points, ceil(roots[end][1][2]) + 1)
+    return points
+end
+
+function _sample_points_0(f::Vector{QQMPolyRingElem})::Vector{Vector{QQFieldElem}}
+    @assert all(map(is_constant, f))
+    return [Vector{QQFieldElem}()]
+end
+
+function _sample_points_1(f::Vector{QQMPolyRingElem}; worker_pool::AbstractWorkerPool=default_worker_pool())::Vector{Vector{QQFieldElem}}
+    @assert all(map(is_univariate, f))
+    R, _ = polynomial_ring(QQ, :x)
+    return [[p] for p in _sample_points(map(p -> to_univariate(R, p), f); worker_pool=worker_pool)]
+end
+
+function _sample_points_desc(n::Int)::String
+    if n == 0
+        return "Constant"
+    elseif n == 1
+        return "Univariate"
+    elseif n == 2
+        return "Bivariate"
+    elseif n == 3
+        return "Trivariate"
+    else
+        return "Multivariate"
+    end
+end
+
+function _sample_points_2(
+    f::Vector{QQMPolyRingElem},
+    xs::Vector{QQMPolyRingElem};
+    nr_thrds::Int=1,
+    worker_pool::AbstractWorkerPool=default_worker_pool(),
+    show_progress::Bool=false,
+    desc::String="$(_sample_points_desc(length(xs))) sample points"
+)::Vector{Vector{QQFieldElem}}
+    @assert length(xs) >= 2
+    x₁ = xs[1:end-1]
+    x₂ = xs[end]
+    factors = unique([p for fₓ in f for (p, _) in factor(fₓ)])
+    v = Vector{QQMPolyRingElem}()
+    for i in eachindex(factors)
+        if !isempty(intersect(x₁, vars(factors[i])))
+            push!(v, leading_coefficient(factors[i], length(xs)))
+        end
+        if x₂ in vars(factors[i])
+            push!(v, Interpolation.discriminant(factors[i], length(xs); nr_thrds=nr_thrds))
+        end
+    end
+    for i in 1:length(factors)-1
+        for j in i+1:length(factors)
+            if x₂ in vars(factors[i]) || x₂ in vars(factors[j])
+                push!(v, Interpolation.resultant(factors[i], factors[j], length(xs); nr_thrds=nr_thrds))
+            end
+        end
+    end
+    p₁ = if length(xs) == 2
+        _sample_points_1(v; worker_pool=worker_pool)
+    else
+        _sample_points_2(v, x₁; show_progress=show_progress, worker_pool=worker_pool)
+    end
+    prog = Progress.ProgressBar(total=length(p₁); desc=desc, enabled=show_progress)
+    Progress.update!(prog, 0)
+    function _points_chunk(p::Vector{Vector{QQFieldElem}})::Vector{Vector{QQFieldElem}}
+        res_chunk = Vector{Vector{QQFieldElem}}()
+        for i in eachindex(p)
+            p₂ = _sample_points_1(map(p′ -> evaluate(p′, x₁, p[i]), f); worker_pool=worker_pool)
+            for j in eachindex(p₂)
+                push!(res_chunk, [p[i]; p₂[j][1]])
+            end
+            Progress.next!(prog)
+        end
+        res_chunk
+    end
+    chunk_size = ceil(Int, length(p₁) / nr_thrds)
+    chunks = [p₁[i:min(i + chunk_size - 1, end)] for i in 1:chunk_size:length(p₁)]
+    tasks = [Threads.@spawn _points_chunk(chunk) for chunk in chunks]
+    res = sort(vcat(fetch.(tasks)...))
+    Progress.finish!(prog)
+    res
+end
+
+@doc Markdown.doc"""
+    _sample_points(f::Vector{QQMPolyRingElem}; nr_thrds::Int=1, worker_pool::AbstractWorkerPool=default_worker_pool(), show_progress::Bool=false)::Vector{Vector{QQFieldElem}}
+
+Given a vector of multivariate polynomials, returns a vector of points such that each connected component of the non-vanishing set of the polynomials contains at least one of the returned points.
+
+**Note**: This is an internal function.
+"""
+function _sample_points(
+    f::Vector{QQMPolyRingElem};
+    nr_thrds::Int=1,
+    worker_pool::AbstractWorkerPool=default_worker_pool(),
+    show_progress::Bool=false
+)::Vector{Vector{QQFieldElem}}
+    if any(is_zero, f)
+        error("Cannot sample points for polynomials containing zero polynomial")
+    end
+    p = Vector{Vector{QQFieldElem}}([])
+    if length(gens(parent(f[1]))) == 0
+        p = _sample_points_0(f)
+    elseif length(gens(parent(f[1]))) == 1
+        p = _sample_points_1(f; worker_pool=worker_pool)
+    else
+        p = _sample_points_2(f, gens(parent(f[1])); nr_thrds=nr_thrds, worker_pool=worker_pool, show_progress=show_progress)
+    end
+    return p
+end
+
+@doc Markdown.doc"""
+    points_per_components(eqs::Vector{QQMPolyRingElem}, pos::Vector{QQMPolyRingElem}, ineqs::Vector{QQMPolyRingElem}; nr_thrds::Int=1, worker_pool::AbstractWorkerPool=default_worker_pool(), info_level::Int=0)::Vector{Vector{Vector{QQFieldElem}}}
+
+Given a semi-algebraic set defined by equations `eqs`, positivity constraints `pos`, and non-vanishing constraints `ineqs`, returns a vector of points meeting all connected components of the set.
+Each point is represented by an isolating box, i.e., a vector of intervals represented by pairs of rational numbers.
+
+**NOTE**: Sampling points per components is currently only implemented for systems without equations, i.e., `eqs` must be empty.
+
+# Arguments
+- `eqs::Vector{QQMPolyRingElem}`: equations defining the semi-algebraic set.
+- `pos::Vector{QQMPolyRingElem}`: positivity constraints defining the semi-algebraic set.
+- `ineqs::Vector{QQMPolyRingElem}`: non-vanishing constraints defining the semi-algebraic set.
+- `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.
+- `info_level::Int=0`: info level printout: off (`0`, default), summary (`1`), detailed (`2`).
+
+# Examples
+
+```jldoctest
+julia> using AlgebraicSolving
+
+julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
+(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
+
+julia> length(points_per_components(QQMPolyRingElem[], [x, y], [x + y - 1])) >= 2
+true
+```
+"""
+function points_per_components(
+    eqs::Vector{QQMPolyRingElem},
+    pos::Vector{QQMPolyRingElem},
+    ineqs::Vector{QQMPolyRingElem};
+    nr_thrds::Int=1,
+    worker_pool::AbstractWorkerPool=default_worker_pool(),
+    info_level::Int=0
+)::Vector{Vector{Vector{QQFieldElem}}}
+    if !isempty(eqs)
+        throw(ArgumentError("Sampling points per components is not implemented for systems involving equations"))
+    end
+    points = _sample_points([pos; ineqs]; nr_thrds=nr_thrds, worker_pool=worker_pool, show_progress=info_level > 0)
+    res = Vector{Vector{Vector{QQFieldElem}}}()
+    for point in points
+        if all(evaluate(constraint, point) > 0 for constraint in pos)
+            push!(res, map(x -> [x, x], point))
+        end
+    end
+    return res
+end
diff --git a/src/exports.jl b/src/exports.jl
index 5e3d57e..726feae 100644
--- a/src/exports.jl
+++ b/src/exports.jl
@@ -3,6 +3,7 @@ export polynomial_ring, MPolyRing, GFElem, MPolyRingElem,
        QQ, vars, nvars, ngens, ZZRingElem, QQFieldElem,
        QQMPolyRingElem, base_ring, coefficient_ring, evaluate,
        prime_field, sig_groebner_basis, cyclic, leading_coefficient,
+       points_per_components,
        equidimensional_decomposition, homogenize, dimension, FqMPolyRingElem,
        hilbert_series, hilbert_dimension, hilbert_degree, hilbert_polynomial,
        rational_curve_parametrization
diff --git a/src/imports.jl b/src/imports.jl
index 632b29a..7a869c3 100644
--- a/src/imports.jl
+++ b/src/imports.jl
@@ -7,7 +7,7 @@ using StaticArrays
 import Random: MersenneTwister
 import Logging: ConsoleLogger, with_logger, Warn, Info
 import Printf: @sprintf, @printf
-import Distributed: AbstractWorkerPool, remotecall_fetch
+import Distributed: AbstractWorkerPool, default_worker_pool, remotecall_fetch
 
 import Nemo:
     bell,
diff --git a/test/algorithms/sampling.jl b/test/algorithms/sampling.jl
new file mode 100644
index 0000000..590dd6e
--- /dev/null
+++ b/test/algorithms/sampling.jl
@@ -0,0 +1,37 @@
+function points_per_components(ineqs::Vector{QQMPolyRingElem})::Vector{Vector{Vector{QQFieldElem}}}
+    AlgebraicSolving.points_per_components(QQMPolyRingElem[], QQMPolyRingElem[], ineqs)
+end
+
+@testset "Algorithms -> Univariate sample points" begin
+    R, (x,) = polynomial_ring(QQ, ["x"])
+    f = [one(R)]
+    @test length(points_per_components(f)) == 1
+    f = [x + 1, x + 4294967295 // 4294967296, x + 4294967297 // 4294967296]
+    @test length(points_per_components(f)) == 4
+end
+
+@testset "Algorithms -> Bivariate sample points" begin
+    R, (x, y) = polynomial_ring(QQ, ["x", "y"])
+    f = [one(R)]
+    @test length(points_per_components(f)) == 1
+    f = [x, x + 1]
+    @test length(points_per_components(f)) == 3
+    f = [y, y + 1]
+    @test length(points_per_components(f)) == 3
+    f = [x, x + 1, y - x, y + x - 1]
+    @test length(points_per_components(f)) >= 9
+end
+
+@testset "Algorithms -> Trivariate sample points" begin
+    R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
+    f = [one(R)]
+    @test length(points_per_components(f)) == 1
+    f = [x, x + 1]
+    @test length(points_per_components(f)) == 3
+    f = [y, y + 1]
+    @test length(points_per_components(f)) == 3
+    f = [x, x + 1, y - x, y + x - 1]
+    @test length(points_per_components(f)) >= 9
+    f = [x, y, z]
+    @test length(points_per_components(f)) == 8
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index f6cda36..052157b 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -10,6 +10,7 @@ include("algorithms/dimension.jl")
 include("algorithms/hilbert.jl")
 include("algorithms/decomposition.jl")
 include("algorithms/param-curves.jl")
+include("algorithms/sampling.jl")
 include("examples/katsura.jl")
 include("interp/thiele.jl") 
 include("interp/newton.jl")