homalg · StrongModuleGroebnerBasis

16 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.StrongModuleGroebnerBasis

A complete normalized strong module Groebner basis over Z.

PythonRust

class StrongModuleGroebnerBasis

noether::homalg::StrongModuleGroebnerBasis · Direct Rust declaration

pub struct StrongModuleGroebnerBasis { /* 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/StrongModuleGroebnerBasis@1

Executable usage: validation/unit/homalg/test_module_facade.py:42 · validation/unit/homalg/test_module_facade.py:721 · validation/unit/homalg/test_module_facade.py:724

homalg.StrongModuleGroebnerBasis.basis

Return the basis recorded by this StrongModuleGroebnerBasis.

Detailed contract

Canonically ordered positive-leading-coefficient strong basis.

PythonRust

(self) -> Submodule[_IntegralModuleRing]

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

pub fn basis(&self) -> &Submodule<PolynomialRing<IntegerRing>>

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/StrongModuleGroebnerBasis@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.StrongModuleGroebnerBasis.contains

Compute contains for this StrongModuleGroebnerBasis.

Detailed contract

Whether an ambient value belongs to the original submodule.

PythonRust

(self, value: ModuleElement[_IntegralModuleRing]) -> bool

noether::homalg::StrongModuleGroebnerBasis::contains · Direct Rust declaration

pub fn contains( &self, value: &ModuleElement<PolynomialRing<IntegerRing>>, ) -> Result<bool, 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/StrongModuleGroebnerBasis@1

Executable usage: examples/38_exact_linear_systems.py:73 · validation/unit/commalg/test_local_algebra.py:63 · validation/unit/commalg/test_prime_localization.py:50 · validation/unit/commalg/test_prime_localization.py:55 · validation/unit/groups/test_acceptance_workload.py:56 · validation/unit/groups/test_permutation_group_facade.py:81

homalg.StrongModuleGroebnerBasis.from_json

Deserialize and revalidate a StrongModuleGroebnerBasis from typed JSON.

Detailed contract

Deserialize and independently recompute the strong basis.

PythonRust

(cls, value: str) -> StrongModuleGroebnerBasis

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

Compute initial submodule for this StrongModuleGroebnerBasis.

Detailed contract

Initial submodule generated by the full coefficient-bearing leading terms.

PythonRust

(self) -> Submodule[_IntegralModuleRing]

noether::homalg::StrongModuleGroebnerBasis::initial_submodule · Direct Rust declaration

pub fn initial_submodule( &self, ) -> Result<Submodule<PolynomialRing<IntegerRing>>, 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/StrongModuleGroebnerBasis@1

Executable usage: validation/unit/homalg/test_module_facade.py:653 · validation/unit/homalg/test_module_facade.py:725

homalg.StrongModuleGroebnerBasis.input

Return the input recorded by this StrongModuleGroebnerBasis.

Detailed contract

Original generated submodule.

PythonRust

(self) -> Submodule[_IntegralModuleRing]

noether::homalg::StrongModuleGroebnerBasis::input · Direct Rust declaration

pub fn input(&self) -> &Submodule<PolynomialRing<IntegerRing>>

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/StrongModuleGroebnerBasis@1

Executable usage: validation/unit/infra/test_mpi_scale_report.py:331 · validation/unit/infra/test_mpi_scale_report.py:363 · validation/unit/infra/test_mpi_scale_report.py:450 · validation/unit/infra/test_mpi_scale_report.py:458 · validation/unit/infra/test_mpi_scale_report.py:469 · validation/unit/infra/test_mpi_scale_report.py:488

homalg.StrongModuleGroebnerBasis.is_complete

Test whether this StrongModuleGroebnerBasis is complete.

Detailed contract

This value is complete; bounded failures do not return partial bases.

PythonRust

(self) -> bool

noether::homalg::StrongModuleGroebnerBasis::is_complete · Direct Rust declaration

pub const fn is_complete(&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/StrongModuleGroebnerBasis@1

Executable usage: validation/unit/homalg/test_module_facade.py:651 · validation/unit/homalg/test_module_facade.py:720 · validation/corpus/homalg/oscar-module-kernel.py:39

homalg.StrongModuleGroebnerBasis.leading_terms

Return the leading terms recorded by this StrongModuleGroebnerBasis.

Detailed contract

Ordered coefficient-bearing leading terms.

PythonRust

(self) -> list[StrongModuleTerm]

noether::homalg::StrongModuleGroebnerBasis::leading_terms · Direct Rust declaration

pub fn leading_terms(&self) -> &[StrongModuleTerm]

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/StrongModuleGroebnerBasis@1

Executable usage: validation/unit/homalg/test_module_facade.py:722 · validation/unit/homalg/test_module_facade.py:723 · validation/unit/homalg/test_module_facade.py:762 · validation/unit/homalg/test_module_facade.py:763

homalg.StrongModuleGroebnerBasis.lift

Compute lift for this StrongModuleGroebnerBasis.

Detailed contract

Original-generator coordinates for a member.

PythonRust

(self, value: ModuleElement[_IntegralModuleRing]) -> ModuleElement[_IntegralModuleRing] | None

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

pub fn lift( &self, value: &ModuleElement<PolynomialRing<IntegerRing>>, ) -> Result<Option<ModuleElement<PolynomialRing<IntegerRing>>>, 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/StrongModuleGroebnerBasis@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.StrongModuleGroebnerBasis.normal_form

Compute normal form for this StrongModuleGroebnerBasis.

Detailed contract

Canonical Euclidean strong normal form.

PythonRust

(self, value: ModuleElement[_IntegralModuleRing]) -> ModuleElement[_IntegralModuleRing]

noether::homalg::StrongModuleGroebnerBasis::normal_form · Direct Rust declaration

pub fn normal_form( &self, value: &ModuleElement<PolynomialRing<IntegerRing>>, ) -> Result<ModuleElement<PolynomialRing<IntegerRing>>, 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/StrongModuleGroebnerBasis@1

Executable usage: validation/unit/commalg/test_graded_commutative_facade.py:26 · validation/unit/commalg/test_integer_groebner_facade.py:17 · validation/unit/commalg/test_polynomial_ideal_facade.py:194 · validation/unit/commalg/test_polynomial_ideal_facade.py:409 · validation/unit/commalg/test_polynomial_ideal_facade.py:441 · validation/unit/commalg/test_polynomial_ideal_facade.py:52

homalg.StrongModuleGroebnerBasis.order

Return the order recorded by this StrongModuleGroebnerBasis.

Detailed contract

Exact module-monomial order.

PythonRust

(self) -> ModuleMonomialOrder

noether::homalg::StrongModuleGroebnerBasis::order · Direct Rust declaration

pub fn order(&self) -> &ModuleMonomialOrder

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/StrongModuleGroebnerBasis@1

Executable usage: examples/11_qadic_witt_vectors.py:14 · examples/12_number_field_ideals.py:13 · examples/12_number_field_ideals.py:14 · examples/12_number_field_ideals.py:17 · examples/15_groebner_twisted_cubic.py:18 · examples/19_permutation_groups.py:13

homalg.StrongModuleGroebnerBasis.reduce

Compute reduce for this StrongModuleGroebnerBasis.

Detailed contract

Reduce an ambient value, retaining canonical-basis quotients and an exact lift into the original generators.

PythonRust

(self, value: ModuleElement[_IntegralModuleRing]) -> StrongModuleReduction

noether::homalg::StrongModuleGroebnerBasis::reduce · Direct Rust declaration

pub fn reduce( &self, value: &ModuleElement<PolynomialRing<IntegerRing>>, ) -> Result<StrongModuleReduction, 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/StrongModuleGroebnerBasis@1

Executable usage: examples/30_modules_and_resolutions.py:28 · validation/unit/commalg/test_local_algebra.py:56 · validation/unit/commalg/test_polynomial_ideal_facade.py:270 · validation/unit/commalg/test_polynomial_ideal_facade.py:304 · validation/unit/homalg/test_local_module.py:28 · validation/unit/homalg/test_module_facade.py:525

homalg.StrongModuleGroebnerBasis.syzygies

Compute syzygies for this StrongModuleGroebnerBasis.

Detailed contract

Complete relation module among the original integral generators.

PythonRust

(self) -> Submodule[_IntegralModuleRing]

noether::homalg::StrongModuleGroebnerBasis::syzygies · Direct Rust declaration

pub fn syzygies( &self, ) -> Result<Submodule<PolynomialRing<IntegerRing>>, 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/StrongModuleGroebnerBasis@1

Executable usage: validation/unit/homalg/test_module_facade.py:738 · validation/unit/homalg/test_module_facade.py:765 · validation/bench/macaulay2_core.py:8 · validation/bench/magma_release.py:1279 · validation/bench/magma_release.py:1292 · validation/bench/magma_release.py:1305

homalg.StrongModuleGroebnerBasis.to_json

Serialize this StrongModuleGroebnerBasis to its canonical typed JSON representation.

Detailed contract

Serialize the complete integral strong-basis evidence.

PythonRust

(self) -> str

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

Return the transformation recorded by this StrongModuleGroebnerBasis.

Detailed contract

Map expressing every strong-basis generator in the original family.

PythonRust

(self) -> ModuleHomomorphism[_IntegralModuleRing]

noether::homalg::StrongModuleGroebnerBasis::transformation · Direct Rust declaration

pub fn transformation(&self) -> &ModuleHomomorphism<PolynomialRing<IntegerRing>>

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/StrongModuleGroebnerBasis@1

Executable usage: examples/04_smith_normal_form.py:3 · validation/unit/linalg/test_lattice_facade.py:172 · validation/unit/linalg/test_lattice_facade.py:173 · validation/unit/linalg/test_lattice_facade.py:179 · validation/bench/macaulay2_core.py:1372 · validation/corpus/linalg/picard-cremona-transformations.py:20

homalg.StrongModuleGroebnerBasis.verify

Replay and verify the mathematical evidence carried by this StrongModuleGroebnerBasis.

Detailed contract

Independently recompute and compare the canonical basis, transformation, and coefficient-bearing leaders.

PythonRust

(self) -> None

noether::homalg::StrongModuleGroebnerBasis::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/StrongModuleGroebnerBasis@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