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commalg · LocalReduction

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

Worked example

The complete checked program Reproduce Fisher's order-five elliptic-function normalization uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Reproduce Fisher's order-five elliptic-function normalization."""

from noether import Q
from noether.commalg import RationalFunctionField, WeilDivisor
from noether.numthy import EllipticCurve, NthPowerClassGroup


def main() -> None:
    coefficients = RationalFunctionField(Q, "lambda")
    parameter = coefficients.generator()
    curve = EllipticCurve(
        coefficients,
        [1 - parameter, -parameter, -parameter, 0, 0],
    )
    point = curve.point(0, 0)
    point.verify_exact_order(5)

    miller = curve.miller_function(point, 5)
    divisor = WeilDivisor(
        curve.underlying_curve,
        [
            (curve.origin().underlying_curve_point.as_closed_point(), 5),
            (point.underlying_curve_point.as_closed_point(), -5),
        ],
    )
    function = divisor.riemann_roch_space().basis[0]
    assert (function / miller.function).as_constant() is not None

    evaluation_at_two = function.evaluate_at((2 * point).underlying_curve_point)
    evaluation_at_four = function.evaluate_at((4 * point).underlying_curve_point)
    assert evaluation_at_two.value is not None
    assert evaluation_at_four.value is not None
    classes = NthPowerClassGroup(coefficients, 5)
    normalization = classes.class_of(evaluation_at_two.value) ** -2 * classes.class_of(
        evaluation_at_four.value
    )
    assert normalization.is_identity()
    print("order 5 verified; Riemann--Roch and Miller agree; normalization is trivial")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/26_elliptic_descent_functions.py.

commalg.LocalReduction

Replayable weak-normal-form evidence for one Mora reduction.

class LocalReduction

noether::commalg::LocalReduction · Direct Rust declaration

pub struct LocalReduction<R: Field> { /* 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/LocalReduction@1

Executable usage: validation/unit/commalg/test_local_algebra.py:13 · validation/unit/commalg/test_local_algebra.py:62

commalg.LocalReduction.basis

Return the basis recorded by this LocalReduction.

Detailed contract

Standard basis used by the reduction.

(self) -> LocalStandardBasis[_ProjectedIdealFieldContext]

noether::commalg::LocalReduction::basis · Direct Rust declaration

pub fn basis(&self) -> &LocalStandardBasis<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/LocalReduction@1

Executable usage: examples/26_elliptic_descent_functions.py:28 · examples/30_modules_and_resolutions.py:27 · examples/30_modules_and_resolutions.py:28 · examples/30_modules_and_resolutions.py:29 · examples/33_simplicial_cohomology_operations.py:26 · examples/33_simplicial_cohomology_operations.py:27

commalg.LocalReduction.from_json

Deserialize and revalidate a LocalReduction from typed JSON.

Detailed contract

Deserialize and exactly replay weak-normal-form evidence.

(cls, value: str) -> LocalReduction[CoefficientRing]

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

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

Test whether this LocalReduction is member.

Detailed contract

Whether the retained remainder is zero.

(self) -> bool

noether::commalg::LocalReduction::is_member · Direct Rust declaration

pub fn is_member(&self) -> bool
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/LocalReduction@1

Executable usage: examples/30_modules_and_resolutions.py:31 · validation/unit/commalg/test_local_algebra.py:58 · validation/unit/commalg/test_polynomial_ideal_facade.py:274 · validation/unit/commalg/test_polynomial_ideal_facade.py:305 · validation/unit/commalg/test_prime_localization.py:45 · validation/unit/commalg/test_prime_localization.py:51

commalg.LocalReduction.multiplier

Return the multiplier recorded by this LocalReduction.

Detailed contract

Unit multiplier in the weak-normal-form identity.

(self) -> LocalPolynomial[_ProjectedIdealFieldContext]

noether::commalg::LocalReduction::multiplier · Direct Rust declaration

pub fn multiplier(&self) -> Result<LocalPolynomial<R>, LocalAlgebraError>
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/LocalReduction@1

Executable usage: validation/unit/commalg/test_local_algebra.py:59 · validation/unit/commalg/test_polynomial_ideal_facade.py:328 · validation/unit/commalg/test_prime_localization.py:46 · validation/unit/commalg/test_prime_localization.py:47 · validation/unit/commalg/test_prime_localization.py:52 · validation/unit/homalg/test_local_module.py:34

commalg.LocalReduction.quotients

Return the quotients recorded by this LocalReduction.

Detailed contract

Coefficients of the ordered standard-basis members.

(self) -> list[LocalPolynomial[_ProjectedIdealFieldContext]]

noether::commalg::LocalReduction::quotients · Direct Rust declaration

pub fn quotients(&self) -> Result<Vec<LocalPolynomial<R>>, LocalAlgebraError>
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/LocalReduction@1

Executable usage: validation/unit/commalg/test_local_algebra.py:60 · validation/unit/commalg/test_prime_localization.py:48 · validation/unit/commalg/test_prime_localization.py:53 · validation/unit/commalg/test_quotient_facade.py:3 · validation/unit/homalg/test_local_module.py:36

commalg.LocalReduction.remainder

Return the remainder recorded by this LocalReduction.

Detailed contract

Weak normal-form remainder.

(self) -> LocalPolynomial[_ProjectedIdealFieldContext]

noether::commalg::LocalReduction::remainder · Direct Rust declaration

pub fn remainder(&self) -> Result<LocalPolynomial<R>, LocalAlgebraError>
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/LocalReduction@1

Executable usage: examples/13_chinese_remainder_theorem.py:3 · validation/unit/commalg/test_polynomial_facade.py:463 · validation/unit/commalg/test_polynomial_facade.py:484 · validation/unit/commalg/test_polynomial_facade.py:501 · validation/unit/commalg/test_polynomial_ideal_facade.py:275 · validation/unit/homalg/test_local_module.py:35

commalg.LocalReduction.to_json

Serialize this LocalReduction to its canonical typed JSON representation.

Detailed contract

Serialize exact weak-normal-form evidence.

(self) -> str

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

pub fn to_json(&self) -> Result<String, NoetherError>where R: Serialize, R::Element: 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/LocalReduction@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.LocalReduction.value

Return the value recorded by this LocalReduction.

Detailed contract

Local element that was reduced.

(self) -> LocalPolynomial[_ProjectedIdealFieldContext]

noether::commalg::LocalReduction::value · Direct Rust declaration

pub fn value(&self) -> &LocalPolynomial<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/LocalReduction@1

Executable usage: examples/09_finite_field_isomorphisms.py:13 · examples/09_finite_field_isomorphisms.py:14 · examples/13_chinese_remainder_theorem.py:12 · examples/13_chinese_remainder_theorem.py:13 · examples/13_chinese_remainder_theorem.py:14 · examples/17_weighted_power_series.py:18

commalg.LocalReduction.verify

Replay and verify the mathematical evidence carried by this LocalReduction.

Detailed contract

Replay the exact weak-normal-form identity and deterministic reduction.

(self) -> None

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

pub fn verify(&self) -> Result<(), LocalAlgebraError>
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/LocalReduction@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