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commalg · LocalStandardBasis

14 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.LocalStandardBasis

A verified Mora standard basis and its local invariants.

class LocalStandardBasis

noether::commalg::LocalStandardBasis · Direct Rust declaration

pub struct LocalStandardBasis<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/LocalStandardBasis@1

Executable usage: validation/unit/commalg/test_local_algebra.py:14 · validation/unit/commalg/test_local_algebra.py:68

commalg.LocalStandardBasis.associated_graded

Compute associated graded for this LocalStandardBasis.

Detailed contract

Associated graded ring of the local quotient.

(self) -> QuotientRing[_ProjectedIdealFieldContext]

noether::commalg::LocalStandardBasis::associated_graded · Direct Rust declaration

pub fn associated_graded( &self, ) -> Result<QuotientRing<PolynomialRing<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/LocalStandardBasis@1

Executable usage: validation/unit/commalg/test_local_algebra.py:66

commalg.LocalStandardBasis.basis

Return the basis recorded by this LocalStandardBasis.

Detailed contract

Deterministic monic Mora standard basis in translated coordinates.

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

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

pub fn basis(&self) -> &[Polynomial<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/LocalStandardBasis@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.LocalStandardBasis.contains

Compute contains for this LocalStandardBasis.

Detailed contract

Local ideal membership, with a replayable weak-normal-form identity.

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

noether::commalg::LocalStandardBasis::contains · Direct Rust declaration

pub fn contains( &self, value: &LocalPolynomial<R>, ) -> Result<bool, 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/LocalStandardBasis@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

commalg.LocalStandardBasis.dimension

Return the dimension recorded by this LocalStandardBasis.

Detailed contract

Krull dimension of the local quotient, computed from its tangent cone.

(self) -> int

noether::commalg::LocalStandardBasis::dimension · Direct Rust declaration

pub fn dimension(&self) -> Result<i64, 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/LocalStandardBasis@1

Executable usage: examples/03_exact_linear_systems.py:15 · examples/03_exact_linear_systems.py:16 · examples/15_groebner_twisted_cubic.py:14 · examples/15_groebner_twisted_cubic.py:28 · examples/27_weight_characters.py:25 · examples/34_finite_group_cohomology.py:13

commalg.LocalStandardBasis.from_json

Deserialize and revalidate a LocalStandardBasis from typed JSON.

Detailed contract

Deserialize and exactly recompute a local standard basis.

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

noether::commalg::LocalStandardBasis::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/LocalStandardBasis@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.LocalStandardBasis.generators

Return the generators recorded by this LocalStandardBasis.

Detailed contract

Canonically ordered numerator generators supplied to the computation.

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

noether::commalg::LocalStandardBasis::generators · Direct Rust declaration

pub fn generators(&self) -> &[Polynomial<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/LocalStandardBasis@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

commalg.LocalStandardBasis.ideal

Return the ideal recorded by this LocalStandardBasis.

Detailed contract

Canonical local ideal represented by this basis.

(self) -> LocalIdeal[_ProjectedIdealFieldContext]

noether::commalg::LocalStandardBasis::ideal · Direct Rust declaration

pub fn ideal(&self) -> LocalIdeal<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/LocalStandardBasis@1

Executable usage: examples/12_number_field_ideals.py:17 · examples/12_number_field_ideals.py:18 · examples/12_number_field_ideals.py:19 · examples/12_number_field_ideals.py:3 · examples/15_groebner_twisted_cubic.py:12 · examples/15_groebner_twisted_cubic.py:13

commalg.LocalStandardBasis.multiplicity

Return the multiplicity recorded by this LocalStandardBasis.

Detailed contract

Hilbert–Samuel multiplicity, computed as the degree of the tangent cone.

(self) -> int

noether::commalg::LocalStandardBasis::multiplicity · Direct Rust declaration

pub fn multiplicity(&self) -> Result<Integer, 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/LocalStandardBasis@1

Executable usage: validation/unit/combin/test_acceptance_workload.py:64 · validation/unit/commalg/test_local_algebra.py:65 · validation/unit/commalg/test_local_algebra.py:82 · validation/unit/commalg/test_polynomial_facade.py:126 · validation/unit/commalg/test_polynomial_facade.py:158 · validation/unit/commalg/test_polynomial_facade.py:96

commalg.LocalStandardBasis.reduce

Compute reduce for this LocalStandardBasis.

Detailed contract

Certified Mora weak normal form with retained coefficients.

(self, value: LocalPolynomial[_ProjectedIdealFieldContext]) -> LocalReduction[_ProjectedIdealFieldContext]

noether::commalg::LocalStandardBasis::reduce · Direct Rust declaration

pub fn reduce( &self, value: &LocalPolynomial<R>, ) -> Result<LocalReduction<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/LocalStandardBasis@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

commalg.LocalStandardBasis.ring

Return the ring recorded by this LocalStandardBasis.

Detailed contract

Exact polynomial local-ring context.

(self) -> LocalPolynomialRing[_ProjectedIdealFieldContext]

noether::commalg::LocalStandardBasis::ring · Direct Rust declaration

pub fn ring(&self) -> &LocalPolynomialRing<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/LocalStandardBasis@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

commalg.LocalStandardBasis.tangent_cone_ideal

Compute tangent cone ideal for this LocalStandardBasis.

Detailed contract

Lowest-degree initial ideal defining the tangent cone.

(self) -> Ideal[PolynomialRing[_ProjectedIdealFieldContext]]

noether::commalg::LocalStandardBasis::tangent_cone_ideal · Direct Rust declaration

pub fn tangent_cone_ideal( &self, ) -> Result<Ideal<PolynomialRing<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/LocalStandardBasis@1

Executable usage: validation/unit/commalg/test_local_algebra.py:83

commalg.LocalStandardBasis.to_json

Serialize this LocalStandardBasis to its canonical typed JSON representation.

Detailed contract

Serialize the context, input generators, and standard basis.

(self) -> str

noether::commalg::LocalStandardBasis::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/LocalStandardBasis@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.LocalStandardBasis.verify

Replay and verify the mathematical evidence carried by this LocalStandardBasis.

Detailed contract

Recompute every Mora pair and compare the retained basis.

(self) -> None

noether::commalg::LocalStandardBasis::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/LocalStandardBasis@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