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commalg · PrimaryDecomposition

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

Worked example

The complete checked program Project and reconstruct a residue class by the Chinese remainder theorem uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Project and reconstruct a residue class by the Chinese remainder theorem."""

from noether.commalg import ResidueRing


def main() -> None:
    ring = ResidueRing(360)
    decomposition = ring.crt()
    decomposition.verify()
    value = ring(347)
    components = decomposition.project(value)
    assert decomposition.reconstruct(components) == value
    print(f"347 mod 360 -> {[int(component) for component in components]}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/13_chinese_remainder_theorem.py.

commalg.PrimaryDecomposition

Complete verified primary decomposition of an admitted polynomial ideal.

class PrimaryDecomposition

noether::commalg::PrimaryDecomposition · Direct Rust declaration

pub struct PrimaryDecomposition<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/PrimaryDecomposition@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1255 · validation/unit/commalg/test_polynomial_ideal_facade.py:1279 · validation/unit/commalg/test_polynomial_ideal_facade.py:1303 · validation/unit/commalg/test_polynomial_ideal_facade.py:1325 · validation/unit/commalg/test_polynomial_ideal_facade.py:29 · validation/unit/commalg/test_polynomial_ideal_facade.py:615

commalg.PrimaryDecomposition.associated_primes

Associated primes, including embedded primes.

Detailed contract

Associated primes, including embedded primes, in component order.

(self) -> list[PolynomialPrimeCertificate[C]]

noether::commalg::PrimaryDecomposition::associated_primes · Direct Rust declaration

pub fn associated_primes(&self) -> Vec<&PolynomialPrimeCertificate<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/PrimaryDecomposition@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1271 · validation/unit/commalg/test_polynomial_ideal_facade.py:566 · validation/unit/commalg/test_polynomial_ideal_facade.py:577 · validation/unit/commalg/test_polynomial_ideal_facade.py:623 · validation/unit/commalg/test_polynomial_ideal_facade.py:654 · validation/unit/commalg/test_polynomial_ideal_facade.py:694

commalg.PrimaryDecomposition.certificates

Exact primary certificate for each component.

Detailed contract

Complete primary evidence in component order.

(self) -> list[PolynomialPrimaryCertificate[C]]

noether::commalg::PrimaryDecomposition::certificates · Direct Rust declaration

pub fn certificates(&self) -> &[PolynomialPrimaryCertificate<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/PrimaryDecomposition@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1021 · validation/unit/commalg/test_polynomial_ideal_facade.py:1089 · validation/unit/commalg/test_polynomial_ideal_facade.py:1090 · validation/unit/commalg/test_polynomial_ideal_facade.py:1261 · validation/unit/commalg/test_polynomial_ideal_facade.py:1276 · validation/unit/commalg/test_polynomial_ideal_facade.py:1293

commalg.PrimaryDecomposition.components

Canonically ordered primary component ideals.

(self) -> list[Ideal[PolynomialRing[C]]]

noether::commalg::PrimaryDecomposition::components · Direct Rust declaration

pub fn components(&self) -> &[Ideal<PolynomialRing<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/PrimaryDecomposition@1

Executable usage: examples/13_chinese_remainder_theorem.py:13 · examples/13_chinese_remainder_theorem.py:14 · examples/13_chinese_remainder_theorem.py:15 · examples/14_weighted_polynomials.py:15 · examples/14_weighted_polynomials.py:16 · examples/14_weighted_polynomials.py:17

commalg.PrimaryDecomposition.from_json

Deserialize and exactly reverify a complete primary decomposition.

(cls, value: str) -> PrimaryDecomposition[RationalField | FiniteField]

noether::commalg::PrimaryDecomposition::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, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/PrimaryDecomposition@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.PrimaryDecomposition.source

Original ideal reconstructed by the primary components.

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

noether::commalg::PrimaryDecomposition::source · Direct Rust declaration

pub fn source(&self) -> &Ideal<PolynomialRing<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/PrimaryDecomposition@1

Executable usage: examples/09_finite_field_isomorphisms.py:11 · examples/09_finite_field_isomorphisms.py:13 · examples/09_finite_field_isomorphisms.py:9 · examples/30_modules_and_resolutions.py:34 · examples/30_modules_and_resolutions.py:37 · validation/unit/category/test_acceptance_workload.py:25

commalg.PrimaryDecomposition.theorem

Exact theorem certifying completeness of the decomposition.

Detailed contract

Exact completeness theorem for the retained component list.

(self) -> str

noether::commalg::PrimaryDecomposition::theorem · Direct Rust declaration

pub const fn theorem(&self) -> &'static str
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/PrimaryDecomposition@1

Executable usage: examples/13_chinese_remainder_theorem.py:3 · validation/unit/commalg/test_polynomial_ideal_facade.py:1150 · validation/unit/commalg/test_polynomial_ideal_facade.py:507 · validation/unit/commalg/test_polynomial_ideal_facade.py:583 · validation/unit/commalg/test_polynomial_ideal_facade.py:659 · validation/unit/homalg/test_module_facade.py:509

commalg.PrimaryDecomposition.to_json

Serialize the complete primary decomposition evidence.

Detailed contract

Serialize the source and complete primary evidence.

(self) -> str

noether::commalg::PrimaryDecomposition::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, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/PrimaryDecomposition@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.PrimaryDecomposition.verify

Recheck every primary claim and the exact component intersection.

Detailed contract

Independently recheck every primary claim and the exact intersection.

(self) -> None

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

pub fn verify(&self) -> Result<(), DecompositionError>
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/PrimaryDecomposition@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