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commalg · FiniteSchemeClosedPointSet

8 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example

The complete checked program Enumerate and certify rational points on a finite-field hyperbola uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Enumerate and certify rational points on a finite-field hyperbola."""

from noether.commalg import AffineSpace
from noether.numthy import FiniteField


def main() -> None:
    field = FiniteField.prime(5)
    plane = AffineSpace(field, 2, names=["x", "y"])
    x, y = plane.coordinate_ring.generators()
    hyperbola = plane.closed_subscheme(plane.coordinate_ring.ideal([x * y - 1]))
    count, certificate = hyperbola.point_count(shards=3)
    certificate.verify(hyperbola)
    assert count == 4
    assert len(list(hyperbola.rational_points())) == count
    print(f"xy=1 over F_5 has {count} rational points")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/18_finite_field_schemes.py.

commalg.FiniteSchemeClosedPointSet

Complete canonical closed-point solution of a finite scheme.

class FiniteSchemeClosedPointSet

noether::commalg::FiniteSchemeClosedPointSet · Direct Rust declaration

pub struct FiniteSchemeClosedPointSet { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/FiniteSchemeClosedPointSet@1

Executable usage: validation/unit/commalg/test_scheme_facade.py:100 · validation/unit/commalg/test_scheme_facade.py:112 · validation/unit/commalg/test_scheme_facade.py:13

commalg.FiniteSchemeClosedPointSet.from_json

Deserialize and revalidate a FiniteSchemeClosedPointSet from typed JSON.

Detailed contract

Deserialize and recompute the complete solution.

(cls, value: str) -> FiniteSchemeClosedPointSet

noether::commalg::FiniteSchemeClosedPointSet::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/FiniteSchemeClosedPointSet@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106 · validation/unit/combin/test_graph_facade.py:25 · validation/unit/combinatorics/test_kashiwara_crystal.py:32

commalg.FiniteSchemeClosedPointSet.geometric_point_count

Return the geometric point count recorded by this FiniteSchemeClosedPointSet.

Detailed contract

Sum of closed-point degrees, without scheme multiplicities.

(self) -> int

noether::commalg::FiniteSchemeClosedPointSet::geometric_point_count · Direct Rust declaration

pub fn geometric_point_count(&self) -> Integer
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/FiniteSchemeClosedPointSet@1

Executable usage: validation/unit/commalg/test_scheme_facade.py:102 · validation/unit/commalg/test_scheme_facade.py:121

commalg.FiniteSchemeClosedPointSet.points

Return the points recorded by this FiniteSchemeClosedPointSet.

Detailed contract

Closed points in deterministic residue-presentation/coordinate order.

(self) -> list[FiniteSchemeClosedPoint]

noether::commalg::FiniteSchemeClosedPointSet::points · Direct Rust declaration

pub fn points(&self) -> &[FiniteSchemeClosedPoint]
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/FiniteSchemeClosedPointSet@1

Executable usage: examples/18_finite_field_schemes.py:18 · examples/18_finite_field_schemes.py:3 · examples/39_kahler_differentials.py:3 · validation/unit/commalg/test_scheme_facade.py:103 · validation/unit/commalg/test_scheme_facade.py:104 · validation/unit/commalg/test_scheme_facade.py:122

commalg.FiniteSchemeClosedPointSet.representation

Return the representation recorded by this FiniteSchemeClosedPointSet.

Detailed contract

RUR used for a nonempty zero-dimensional quotient.

(self) -> RationalUnivariateRepresentation[FiniteField] | None

noether::commalg::FiniteSchemeClosedPointSet::representation · Direct Rust declaration

pub fn representation( &self, ) -> Option<&RationalUnivariateRepresentation<FiniteField>>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/FiniteSchemeClosedPointSet@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:100 · validation/unit/commalg/test_polynomial_ideal_facade.py:101 · validation/unit/commalg/test_polynomial_ideal_facade.py:102 · validation/unit/commalg/test_polynomial_ideal_facade.py:103 · validation/unit/commalg/test_polynomial_ideal_facade.py:104 · validation/unit/commalg/test_polynomial_ideal_facade.py:105

commalg.FiniteSchemeClosedPointSet.scheme

Return the scheme recorded by this FiniteSchemeClosedPointSet.

Detailed contract

Solved affine finite-field scheme.

(self) -> Scheme[FiniteField]

noether::commalg::FiniteSchemeClosedPointSet::scheme · Direct Rust declaration

pub fn scheme(&self) -> &Scheme<FiniteField>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/FiniteSchemeClosedPointSet@1

Executable usage: validation/unit/commalg/test_base_change.py:105 · validation/unit/commalg/test_base_change.py:106 · validation/unit/commalg/test_base_change.py:113 · validation/unit/commalg/test_base_change.py:3 · validation/unit/commalg/test_base_change.py:57 · validation/unit/commalg/test_scheme_facade.py:101

commalg.FiniteSchemeClosedPointSet.to_json

Serialize this FiniteSchemeClosedPointSet to its canonical typed JSON representation.

Detailed contract

Serialize the scheme, RUR, and all verified closed points.

(self) -> str

noether::commalg::FiniteSchemeClosedPointSet::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/FiniteSchemeClosedPointSet@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/category/test_category.py:15 · validation/unit/combin/test_acceptance_workload.py:30 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106

commalg.FiniteSchemeClosedPointSet.verify

Replay and verify the mathematical evidence carried by this FiniteSchemeClosedPointSet.

Detailed contract

Recompute the RUR solution and compare every canonical closed point.

(self) -> None

noether::commalg::FiniteSchemeClosedPointSet::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), SchemeError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/FiniteSchemeClosedPointSet@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:3 · examples/03_exact_linear_systems.py:13 · examples/03_exact_linear_systems.py:3 · examples/04_smith_normal_form.py:12 · examples/06_shortest_lattice_vectors.py:12