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commalg · EquidimensionalRadicalComponent

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

Worked example

The complete checked program Solve an underdetermined rational system and verify its affine space uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Solve an underdetermined rational system and verify its affine space."""

from noether import Q
from noether.linalg import Matrix, Vector


def main() -> None:
    matrix = Matrix.from_rows(Q, [[1, 2, 3], [2, 4, 6]])
    right_hand_side = Vector.from_values(Q, [1, 2])
    solutions = matrix.solve(right_hand_side)
    solutions.verify()
    assert matrix @ solutions.particular == right_hand_side
    assert solutions.dimension == 2
    print(f"particular={solutions.particular}; nullity={solutions.dimension}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/03_exact_linear_systems.py.

commalg.EquidimensionalRadicalComponent

One verified equidimensional radical component.

class EquidimensionalRadicalComponent

noether::commalg::EquidimensionalRadicalComponent · Direct Rust declaration

pub struct EquidimensionalRadicalComponent<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/EquidimensionalRadicalComponent@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1233 · validation/unit/commalg/test_polynomial_ideal_facade.py:1242 · validation/unit/commalg/test_polynomial_ideal_facade.py:17

commalg.EquidimensionalRadicalComponent.dimension

Return the dimension recorded by this EquidimensionalRadicalComponent.

Detailed contract

Common Krull dimension of every minimal prime in this component.

(self) -> int

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

pub const fn dimension(&self) -> i64
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/EquidimensionalRadicalComponent@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.EquidimensionalRadicalComponent.from_json

Deserialize and revalidate a EquidimensionalRadicalComponent from typed JSON.

Detailed contract

Deserialize and exactly reverify equidimensional component evidence.

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

noether::commalg::EquidimensionalRadicalComponent::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/EquidimensionalRadicalComponent@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.EquidimensionalRadicalComponent.ideal

Return the ideal recorded by this EquidimensionalRadicalComponent.

Detailed contract

Intersection of the component’s minimal primes.

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

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

pub fn ideal(&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/EquidimensionalRadicalComponent@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.EquidimensionalRadicalComponent.prime_certificates

Return the prime certificates recorded by this EquidimensionalRadicalComponent.

Detailed contract

Canonically ordered exact primality evidence for this component.

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

noether::commalg::EquidimensionalRadicalComponent::prime_certificates · Direct Rust declaration

pub fn prime_certificates(&self) -> &[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/EquidimensionalRadicalComponent@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1006 · validation/unit/commalg/test_polynomial_ideal_facade.py:1041 · validation/unit/commalg/test_polynomial_ideal_facade.py:1057 · validation/unit/commalg/test_polynomial_ideal_facade.py:1075 · validation/unit/commalg/test_polynomial_ideal_facade.py:1140 · validation/unit/commalg/test_polynomial_ideal_facade.py:1143

commalg.EquidimensionalRadicalComponent.theorem

Return the theorem recorded by this EquidimensionalRadicalComponent.

Detailed contract

Exact theorem used to certify equidimensionality.

(self) -> str

noether::commalg::EquidimensionalRadicalComponent::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/EquidimensionalRadicalComponent@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.EquidimensionalRadicalComponent.to_json

Serialize this EquidimensionalRadicalComponent to its canonical typed JSON representation.

Detailed contract

Serialize the component and complete prime evidence.

(self) -> str

noether::commalg::EquidimensionalRadicalComponent::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/EquidimensionalRadicalComponent@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.EquidimensionalRadicalComponent.verify

Replay and verify the mathematical evidence carried by this EquidimensionalRadicalComponent.

Detailed contract

Independently recheck the equidimensional component evidence.

(self) -> None

noether::commalg::EquidimensionalRadicalComponent::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/EquidimensionalRadicalComponent@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