homalg · LocalModuleStandardBasis

12 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.LocalModuleStandardBasis

A verified Mora standard basis of a submodule over a polynomial local ring.

PythonRust

class LocalModuleStandardBasis

noether::homalg::LocalModuleStandardBasis · Direct Rust declaration

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

Executable usage: validation/unit/homalg/test_local_module.py:22 · validation/unit/homalg/test_local_module.py:41 · validation/unit/homalg/test_local_module.py:53 · validation/unit/homalg/test_local_module.py:9

homalg.LocalModuleStandardBasis.basis

Return the basis recorded by this LocalModuleStandardBasis.

Detailed contract

Deterministic monic Mora standard basis as ambient local elements.

PythonRust

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

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

pub fn basis( &self, ) -> Result<Vec<ModuleElement<LocalPolynomialRing<R>>>, LocalModuleError>

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/LocalModuleStandardBasis@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.LocalModuleStandardBasis.contains

Compute contains for this LocalModuleStandardBasis.

Detailed contract

Decide submodule membership with retained weak-normal-form evidence.

PythonRust

(self, value: Sequence[LocalPolynomial[_LocalModuleField]]) -> bool

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

pub fn contains( &self, value: &ModuleElement<LocalPolynomialRing<R>>, ) -> Result<bool, LocalModuleError>

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/LocalModuleStandardBasis@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.LocalModuleStandardBasis.from_generators

Construct a LocalModuleStandardBasis using generators.

Detailed contract

Compute a local module standard basis from exact ambient elements.

PythonRust

(cls, ring: LocalPolynomialRing[_LocalModuleField], rank: int, generators: Sequence[Sequence[LocalPolynomial[_LocalModuleField]]], *, order: LocalModuleOrder | None=None) -> LocalModuleStandardBasis[_LocalModuleField]

noether::homalg::LocalModuleStandardBasis::from_generators · Direct Rust declaration

pub fn from_generators( ambient: FreeModule<LocalPolynomialRing<R>>, generators: &[ModuleElement<LocalPolynomialRing<R>>], order: LocalModuleOrder, ) -> Result<Self, LocalModuleError>

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/LocalModuleStandardBasis@1

Executable usage: examples/19_permutation_groups.py:9 · examples/20_conjugacy_classes.py:9 · examples/30_modules_and_resolutions.py:26 · examples/34_finite_group_cohomology.py:10 · examples/38_exact_linear_systems.py:56 · validation/unit/groups/test_acceptance_workload.py:122

homalg.LocalModuleStandardBasis.from_json

Deserialize and revalidate a LocalModuleStandardBasis from typed JSON.

Detailed contract

Deserialize and exactly recompute a local module standard basis.

PythonRust

(cls, value: str) -> LocalModuleStandardBasis[_LocalModuleField]

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

Return the generators recorded by this LocalModuleStandardBasis.

Detailed contract

Unit-cleared input generators as ambient local elements.

PythonRust

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

noether::homalg::LocalModuleStandardBasis::generators · Direct Rust declaration

pub fn generators( &self, ) -> Result<Vec<ModuleElement<LocalPolynomialRing<R>>>, LocalModuleError>

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/LocalModuleStandardBasis@1

Executable usage: examples/14_weighted_polynomials.py:11 · examples/15_groebner_twisted_cubic.py:11 · examples/15_groebner_twisted_cubic.py:19 · examples/17_weighted_power_series.py:15 · examples/18_finite_field_schemes.py:12 · examples/30_modules_and_resolutions.py:20

homalg.LocalModuleStandardBasis.order

Return the order recorded by this LocalModuleStandardBasis.

Detailed contract

Module-monomial order extending the local order of the coefficient ring.

PythonRust

(self) -> LocalModuleOrder

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

pub const fn order(&self) -> LocalModuleOrder

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/LocalModuleStandardBasis@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.LocalModuleStandardBasis.rank

Return the rank recorded by this LocalModuleStandardBasis.

PythonRust

(self) -> int

noether::homalg::LocalModuleStandardBasis · Python projection

Python projection convenience over this Rust mathematical type

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/LocalModuleStandardBasis@1

Executable usage: examples/38_exact_linear_systems.py:78 · examples/distributed/f2_linear_algebra.py:34 · examples/distributed/f2_linear_algebra.py:54 · examples/distributed/f2_linear_algebra.py:55 · examples/distributed/f2_linear_algebra.py:60 · validation/unit/combin/test_acceptance_workload.py:74

homalg.LocalModuleStandardBasis.reduce

Compute reduce for this LocalModuleStandardBasis.

Detailed contract

Certified Mora weak normal form of one ambient local element.

PythonRust

(self, value: Sequence[LocalPolynomial[_LocalModuleField]]) -> LocalModuleReduction[_LocalModuleField]

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

pub fn reduce( &self, value: &ModuleElement<LocalPolynomialRing<R>>, ) -> Result<LocalModuleReduction<R>, LocalModuleError>

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/LocalModuleStandardBasis@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.LocalModuleStandardBasis.ring

Return the ring recorded by this LocalModuleStandardBasis.

PythonRust

(self) -> LocalPolynomialRing[_LocalModuleField]

noether::homalg::LocalModuleStandardBasis · Python projection

Python projection convenience over this Rust mathematical type

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/LocalModuleStandardBasis@1

Executable usage: examples/11_qadic_witt_vectors.py:3 · examples/13_chinese_remainder_theorem.py:10 · examples/13_chinese_remainder_theorem.py:12 · examples/13_chinese_remainder_theorem.py:9 · examples/14_weighted_polynomials.py:10 · examples/14_weighted_polynomials.py:11

homalg.LocalModuleStandardBasis.to_json

Serialize this LocalModuleStandardBasis to its canonical typed JSON representation.

Detailed contract

Serialize the ambient module, order, inputs, and standard basis.

PythonRust

(self) -> str

noether::homalg::LocalModuleStandardBasis::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/LocalModuleStandardBasis@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.LocalModuleStandardBasis.verify

Replay and verify the mathematical evidence carried by this LocalModuleStandardBasis.

Detailed contract

Recompute every Mora pair and compare the retained standard basis.

PythonRust

(self) -> None

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

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

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/LocalModuleStandardBasis@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