homalg · ModuleReduction

11 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.

homalg.ModuleReduction

Exact quotient, remainder, and original-generator lift evidence.

PythonRust

class ModuleReduction

noether::homalg::ModuleReduction · Direct Rust declaration

pub struct ModuleReduction<R: Ring> { /* private fields */ }

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@1

Executable usage: validation/unit/homalg/test_module_facade.py:36 · validation/unit/homalg/test_module_facade.py:531 · validation/unit/homalg/test_module_facade.py:668

homalg.ModuleReduction.basis

Return the basis recorded by this ModuleReduction.

Detailed contract

Groebner basis used for this reduction.

PythonRust

(self) -> ModuleGroebnerBasis[_GroebnerModuleRing]

noether::homalg::ModuleReduction::basis · Direct Rust declaration

pub fn basis(&self) -> &ModuleGroebnerBasis<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@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

homalg.ModuleReduction.from_json

Deserialize and revalidate a ModuleReduction from typed JSON.

Detailed contract

Deserialize and independently verify normal-form and lift evidence.

PythonRust

(cls, value: str) -> ModuleReduction[_GroebnerModuleRing]

noether::homalg::ModuleReduction::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, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@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

homalg.ModuleReduction.is_member

Test whether this ModuleReduction is member.

Detailed contract

Whether this reduction proves input-submodule membership.

PythonRust

(self) -> bool

noether::homalg::ModuleReduction::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, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@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

homalg.ModuleReduction.lift

Compute lift for this ModuleReduction.

Detailed contract

Original-generator lift when this reduction proves membership.

PythonRust

(self) -> ModuleElement[_GroebnerModuleRing] | None

noether::homalg::ModuleReduction::lift · Direct Rust declaration

pub fn lift(&self) -> Option<&ModuleElement<R>>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@1

Executable usage: examples/11_qadic_witt_vectors.py:12 · examples/11_qadic_witt_vectors.py:13 · examples/11_qadic_witt_vectors.py:14 · examples/11_qadic_witt_vectors.py:15 · validation/unit/commalg/test_acceptance_workload.py:33 · validation/unit/commalg/test_acceptance_workload.py:34

homalg.ModuleReduction.original_combination

Return the original combination recorded by this ModuleReduction.

Detailed contract

Combination of the original input generators contributing to value.

PythonRust

(self) -> ModuleElement[_GroebnerModuleRing]

noether::homalg::ModuleReduction::original_combination · Direct Rust declaration

pub fn original_combination(&self) -> &ModuleElement<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@1

homalg.ModuleReduction.quotient

Return the quotient recorded by this ModuleReduction.

Detailed contract

Quotient coordinates in the canonical basis generators.

PythonRust

(self) -> ModuleElement[_GroebnerModuleRing]

noether::homalg::ModuleReduction::quotient · Direct Rust declaration

pub fn quotient(&self) -> &ModuleElement<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@1

Executable usage: examples/15_groebner_twisted_cubic.py:24 · examples/15_groebner_twisted_cubic.py:25 · examples/39_kahler_differentials.py:13 · validation/unit/commalg/test_kahler_differentials.py:109 · validation/unit/commalg/test_kahler_differentials.py:128 · validation/unit/commalg/test_kahler_differentials.py:56

homalg.ModuleReduction.remainder

Return the remainder recorded by this ModuleReduction.

Detailed contract

Canonical module remainder.

PythonRust

(self) -> ModuleElement[_GroebnerModuleRing]

noether::homalg::ModuleReduction::remainder · Direct Rust declaration

pub fn remainder(&self) -> &ModuleElement<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@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

homalg.ModuleReduction.to_json

Serialize this ModuleReduction to its canonical typed JSON representation.

Detailed contract

Serialize the complete normal-form and lift evidence as typed JSON.

PythonRust

(self) -> str

noether::homalg::ModuleReduction::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, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@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

homalg.ModuleReduction.value

Return the value recorded by this ModuleReduction.

Detailed contract

Ambient value that was reduced.

PythonRust

(self) -> ModuleElement[_GroebnerModuleRing]

noether::homalg::ModuleReduction::value · Direct Rust declaration

pub fn value(&self) -> &ModuleElement<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@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

homalg.ModuleReduction.verify

Replay and verify the mathematical evidence carried by this ModuleReduction.

Detailed contract

Verify both exact reconstruction identities.

PythonRust

(self) -> None

noether::homalg::ModuleReduction::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), ModuleGroebnerError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded
Semantic ID	noether/homalg/ModuleReduction@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