commalg · MinimalPrimeDecomposition

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.MinimalPrimeDecomposition

Verified minimal primes of an admitted polynomial ideal.

PythonRust

class MinimalPrimeDecomposition

noether::commalg::MinimalPrimeDecomposition · Direct Rust declaration

pub struct MinimalPrimeDecomposition<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/MinimalPrimeDecomposition@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1136 · validation/unit/commalg/test_polynomial_ideal_facade.py:1153 · validation/unit/commalg/test_polynomial_ideal_facade.py:1159 · validation/unit/commalg/test_polynomial_ideal_facade.py:24

commalg.MinimalPrimeDecomposition.components

Canonically ordered minimal prime ideals.

PythonRust

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

noether::commalg::MinimalPrimeDecomposition::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/MinimalPrimeDecomposition@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.MinimalPrimeDecomposition.from_json

Deserialize and exactly reverify minimal-prime evidence.

Detailed contract

Deserialize and exactly reverify rational minimal-prime evidence. Deserialize and exactly reverify finite-field minimal-prime evidence.

PythonRust

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

noether::commalg::MinimalPrimeDecomposition::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/MinimalPrimeDecomposition@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.MinimalPrimeDecomposition.prime_certificates

Exact primality certificate for each component.

Detailed contract

Exact primality certificate for each component, in component order.

PythonRust

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

noether::commalg::MinimalPrimeDecomposition::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/MinimalPrimeDecomposition@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.MinimalPrimeDecomposition.radical

Verified radical of the source ideal.

PythonRust

(self) -> RadicalIdeal[C]

noether::commalg::MinimalPrimeDecomposition::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/MinimalPrimeDecomposition@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.MinimalPrimeDecomposition.source

Original ideal whose minimal primes are represented.

PythonRust

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

noether::commalg::MinimalPrimeDecomposition::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/MinimalPrimeDecomposition@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.MinimalPrimeDecomposition.theorem

Exact theorem certifying the recorded components as prime.

Detailed contract

Exact completeness theorem for the certified component list.

PythonRust

(self) -> str

noether::commalg::MinimalPrimeDecomposition::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/MinimalPrimeDecomposition@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.MinimalPrimeDecomposition.to_json

Serialize the complete minimal-prime evidence.

Detailed contract

Serialize the complete radical, component, and prime evidence.

PythonRust

(self) -> str

noether::commalg::MinimalPrimeDecomposition::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/MinimalPrimeDecomposition@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.MinimalPrimeDecomposition.verify

Recheck radical equality, component intersection, and primality evidence.

Detailed contract

Recheck radical equality, component intersection, and every prime certificate.

PythonRust

(self) -> None

noether::commalg::MinimalPrimeDecomposition::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/MinimalPrimeDecomposition@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