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commalg · FiniteSchemeClosedPoint

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

Worked example

The complete checked program Decompose a polynomial by a nonstandard positive grading uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Decompose a polynomial by a nonstandard positive grading."""

from noether import Z
from noether.commalg import PolynomialRing, WeightedGrading


def main() -> None:
    ring = PolynomialRing(Z, ["x", "y", "z"], grading=WeightedGrading([1, 2, 3]))
    x, y, z = ring.generators()
    sparse = ring.from_terms([([0, 0, 0], 1), ([2, 0, 0], 1), ([0, 1, 0], 1)])
    polynomial = sparse + z
    assert sparse * (x + y) == (1 + x**2 + y) * (x + y)
    components = polynomial.homogeneous_components()
    assert [degree for degree, _ in components] == [0, 2, 3]
    assert x**2 + y == components[1][1]
    print(f"weighted homogeneous degrees: {[degree for degree, _ in components]}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/14_weighted_polynomials.py.

commalg.FiniteSchemeClosedPoint

One canonical closed point with an explicit finite residue field.

class FiniteSchemeClosedPoint

noether::commalg::FiniteSchemeClosedPoint · Direct Rust declaration

pub struct FiniteSchemeClosedPoint { /* 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/FiniteSchemeClosedPoint@1

Executable usage: validation/unit/commalg/test_scheme_facade.py:105 · validation/unit/commalg/test_scheme_facade.py:111 · validation/unit/commalg/test_scheme_facade.py:12

commalg.FiniteSchemeClosedPoint.base_embedding

Return the base embedding recorded by this FiniteSchemeClosedPoint.

Detailed contract

Checked embedding of the base field into the residue field.

(self) -> FiniteFieldEmbedding

noether::commalg::FiniteSchemeClosedPoint::base_embedding · Direct Rust declaration

pub fn base_embedding(&self) -> &FiniteFieldEmbedding
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/FiniteSchemeClosedPoint@1

commalg.FiniteSchemeClosedPoint.coordinates

Return the coordinates recorded by this FiniteSchemeClosedPoint.

Detailed contract

Canonical geometric coordinates in the residue field.

(self) -> list[FiniteFieldElement]

noether::commalg::FiniteSchemeClosedPoint::coordinates · Direct Rust declaration

pub fn coordinates(&self) -> &[FiniteFieldElement]
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/FiniteSchemeClosedPoint@1

Executable usage: validation/unit/commalg/test_base_change.py:114 · validation/unit/commalg/test_base_change.py:54 · validation/unit/commalg/test_base_change.py:62 · validation/unit/commalg/test_kahler_differentials.py:131 · validation/unit/commalg/test_polynomial_ideal_facade.py:171 · validation/unit/commalg/test_quotient_facade.py:142

commalg.FiniteSchemeClosedPoint.degree

Return the degree recorded by this FiniteSchemeClosedPoint.

Detailed contract

Degree of the closed point over the scheme’s base field.

(self) -> int

noether::commalg::FiniteSchemeClosedPoint::degree · Direct Rust declaration

pub fn degree(&self) -> usize
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/FiniteSchemeClosedPoint@1

Executable usage: examples/14_weighted_polynomials.py:16 · examples/14_weighted_polynomials.py:18 · examples/15_groebner_twisted_cubic.py:15 · examples/15_groebner_twisted_cubic.py:28 · examples/31_local_fields_and_precision.py:37 · examples/33_simplicial_cohomology_operations.py:26

commalg.FiniteSchemeClosedPoint.frobenius_orbit

Return the frobenius orbit recorded by this FiniteSchemeClosedPoint.

Detailed contract

Complete relative-Frobenius orbit, starting at the canonical conjugate.

(self) -> list[list[FiniteFieldElement]]

noether::commalg::FiniteSchemeClosedPoint::frobenius_orbit · Direct Rust declaration

pub fn frobenius_orbit(&self) -> &[Vec<FiniteFieldElement>]
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/FiniteSchemeClosedPoint@1

Executable usage: validation/unit/commalg/test_scheme_facade.py:108

commalg.FiniteSchemeClosedPoint.from_json

Deserialize and revalidate a FiniteSchemeClosedPoint from typed JSON.

Detailed contract

Deserialize and exactly reverify a closed point.

(cls, value: str) -> FiniteSchemeClosedPoint

noether::commalg::FiniteSchemeClosedPoint::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/FiniteSchemeClosedPoint@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.FiniteSchemeClosedPoint.residue_field

Return the residue field recorded by this FiniteSchemeClosedPoint.

Detailed contract

Explicit finite residue-field presentation.

(self) -> FiniteField

noether::commalg::FiniteSchemeClosedPoint::residue_field · Direct Rust declaration

pub fn residue_field(&self) -> &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/FiniteSchemeClosedPoint@1

Executable usage: examples/10_padic_hensel_lifting.py:14 · examples/11_qadic_witt_vectors.py:10 · examples/11_qadic_witt_vectors.py:11 · examples/11_qadic_witt_vectors.py:14 · examples/11_qadic_witt_vectors.py:9 · validation/unit/commalg/test_acceptance_workload.py:25

commalg.FiniteSchemeClosedPoint.scheme

Return the scheme recorded by this FiniteSchemeClosedPoint.

Detailed contract

Scheme containing this closed point.

(self) -> Scheme[FiniteField]

noether::commalg::FiniteSchemeClosedPoint::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/FiniteSchemeClosedPoint@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.FiniteSchemeClosedPoint.to_json

Serialize this FiniteSchemeClosedPoint to its canonical typed JSON representation.

Detailed contract

Serialize the complete residue-field and orbit evidence.

(self) -> str

noether::commalg::FiniteSchemeClosedPoint::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/FiniteSchemeClosedPoint@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.FiniteSchemeClosedPoint.verify

Replay and verify the mathematical evidence carried by this FiniteSchemeClosedPoint.

Detailed contract

Recheck the residue presentation, orbit, minimal degree, and equations.

(self) -> None

noether::commalg::FiniteSchemeClosedPoint::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/FiniteSchemeClosedPoint@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