Skip to content

commalg · SchemePoint

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

Worked example

The complete checked program Compute and verify a Bézout certificate over the integers uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Compute and verify a Bézout certificate over the integers."""

from noether import Z


def main() -> None:
    identity = Z.extended_gcd(987_654_321, 123_456_789)
    identity.verify(Z, 987_654_321, 123_456_789)
    assert (
        identity.coefficient_a * 987_654_321 + identity.coefficient_b * 123_456_789 == identity.gcd
    )
    print(identity)


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/01_bezout_certificates.py.

commalg.SchemePoint

A checked rational point on one finite-field scheme.

class SchemePoint

noether::commalg::SchemePoint · Direct Rust declaration

pub struct SchemePoint<R: CommutativeRing> { /* 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/SchemePoint@1

Executable usage: validation/unit/commalg/test_scheme_facade.py:19 · validation/unit/commalg/test_scheme_facade.py:77

commalg.SchemePoint.base_change

Extend this point and its complete scheme presentation to the target finite field.

Detailed contract

Extend this point and its complete scheme presentation to the target finite field.

Extend the point and its complete scheme presentation along a checked field map.

(self, map: PrimeSubfieldEmbedding | FiniteFieldEmbedding) -> SchemePoint

noether::commalg::SchemePoint::base_change · Direct Rust declaration

pub fn base_change<M>( &self, map: &M, ) -> Result<SchemePoint<M::Target>, BaseChangeError>where M: CoefficientMap<Source = R>, M::Target: Field,
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/SchemePoint@1

Executable usage: validation/unit/commalg/test_base_change.py:108 · validation/unit/commalg/test_base_change.py:110 · validation/unit/commalg/test_base_change.py:111 · validation/unit/commalg/test_base_change.py:112 · validation/unit/commalg/test_base_change.py:113 · validation/unit/commalg/test_base_change.py:114

commalg.SchemePoint.coordinates

Return the coordinates recorded by this SchemePoint.

Detailed contract

Canonical affine or normalized projective coordinates.

(self) -> list[FiniteFieldElement]

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

pub fn coordinates(&self) -> &[R::Element]
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/SchemePoint@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.SchemePoint.from_json

Deserialize and revalidate a SchemePoint from typed JSON.

Detailed contract

Restore and recheck this point.

(cls, value: str) -> SchemePoint

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

pub fn from_json(value: &str) -> Result<Self, NoetherError>where Self: for<'de> Deserialize<'de>,
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/SchemePoint@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.SchemePoint.scheme

Return the scheme recorded by this SchemePoint.

Detailed contract

Target scheme.

(self) -> Scheme[FiniteField]

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

pub fn scheme(&self) -> &Scheme<R>
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/SchemePoint@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.SchemePoint.to_json

Serialize this SchemePoint to its canonical typed JSON representation.

Detailed contract

Serialize this checked point.

(self) -> str

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

pub fn to_json(&self) -> Result<String, NoetherError>where Self: Serialize,
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/SchemePoint@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.SchemePoint.verify

Replay and verify the mathematical evidence carried by this SchemePoint.

Detailed contract

Re-evaluate every defining equation and the projective normalization.

(self) -> None

noether::commalg::SchemePoint::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/SchemePoint@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