commalg · EquidimensionalRadicalDecomposition

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.EquidimensionalRadicalDecomposition

Verified decomposition of a polynomial-ideal radical by dimension.

PythonRust

class EquidimensionalRadicalDecomposition

noether::commalg::EquidimensionalRadicalDecomposition · Direct Rust declaration

pub struct EquidimensionalRadicalDecomposition<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/EquidimensionalRadicalDecomposition@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1228 · validation/unit/commalg/test_polynomial_ideal_facade.py:1244 · validation/unit/commalg/test_polynomial_ideal_facade.py:18

commalg.EquidimensionalRadicalDecomposition.components

Return the components recorded by this EquidimensionalRadicalDecomposition.

Detailed contract

Components in strictly decreasing dimension order.

PythonRust

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

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

pub fn components(&self) -> &[EquidimensionalRadicalComponent<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/EquidimensionalRadicalDecomposition@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.EquidimensionalRadicalDecomposition.from_json

Deserialize and revalidate a EquidimensionalRadicalDecomposition from typed JSON.

Detailed contract

Deserialize and exactly reverify the complete decomposition.

PythonRust

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

noether::commalg::EquidimensionalRadicalDecomposition::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/EquidimensionalRadicalDecomposition@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.EquidimensionalRadicalDecomposition.minimal_primes

Return the minimal primes recorded by this EquidimensionalRadicalDecomposition.

Detailed contract

Complete minimal-prime evidence underlying the grouping.

PythonRust

(self) -> MinimalPrimeDecomposition[C]

noether::commalg::EquidimensionalRadicalDecomposition::minimal_primes · Direct Rust declaration

pub fn minimal_primes(&self) -> &MinimalPrimeDecomposition<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/EquidimensionalRadicalDecomposition@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:69 · validation/unit/commalg/test_acceptance_workload.py:70 · validation/unit/commalg/test_acceptance_workload.py:75 · validation/unit/commalg/test_affine_normalization_facade.py:183 · validation/unit/commalg/test_polynomial_ideal_facade.py:1002 · validation/unit/commalg/test_polynomial_ideal_facade.py:1039

commalg.EquidimensionalRadicalDecomposition.radical

Return the radical recorded by this EquidimensionalRadicalDecomposition.

Detailed contract

Verified radical reconstructed by the equidimensional components.

PythonRust

(self) -> RadicalIdeal[C]

noether::commalg::EquidimensionalRadicalDecomposition::radical · Direct Rust declaration

pub fn radical(&self) -> &RadicalIdeal<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/EquidimensionalRadicalDecomposition@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:35 · validation/unit/commalg/test_polynomial_ideal_facade.py:1000 · validation/unit/commalg/test_polynomial_ideal_facade.py:1023 · validation/unit/commalg/test_polynomial_ideal_facade.py:1034 · validation/unit/commalg/test_polynomial_ideal_facade.py:1035 · validation/unit/commalg/test_polynomial_ideal_facade.py:1036

commalg.EquidimensionalRadicalDecomposition.source

Return the source recorded by this EquidimensionalRadicalDecomposition.

Detailed contract

Original ideal before passage to its radical.

PythonRust

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

noether::commalg::EquidimensionalRadicalDecomposition::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/EquidimensionalRadicalDecomposition@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.EquidimensionalRadicalDecomposition.theorem

Return the theorem recorded by this EquidimensionalRadicalDecomposition.

Detailed contract

Exact completeness theorem for the grouping.

PythonRust

(self) -> str

noether::commalg::EquidimensionalRadicalDecomposition::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/EquidimensionalRadicalDecomposition@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.EquidimensionalRadicalDecomposition.to_json

Serialize this EquidimensionalRadicalDecomposition to its canonical typed JSON representation.

Detailed contract

Serialize the minimal-prime evidence and canonical grouping.

PythonRust

(self) -> str

noether::commalg::EquidimensionalRadicalDecomposition::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/EquidimensionalRadicalDecomposition@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.EquidimensionalRadicalDecomposition.verify

Replay and verify the mathematical evidence carried by this EquidimensionalRadicalDecomposition.

Detailed contract

Independently recheck the grouping and radical reconstruction.

PythonRust

(self) -> None

noether::commalg::EquidimensionalRadicalDecomposition::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/EquidimensionalRadicalDecomposition@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