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commalg · ArithmeticFiberSupport

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

Worked example

The complete checked program Compute and verify a Bézout certificate over the integers uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Compute and verify a Bézout certificate over the integers."""

from noether import Z


def main() -> None:
    identity = Z.extended_gcd(987_654_321, 123_456_789)
    identity.verify(Z, 987_654_321, 123_456_789)
    assert (
        identity.coefficient_a * 987_654_321 + identity.coefficient_b * 123_456_789 == identity.gcd
    )
    print(identity)


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/01_bezout_certificates.py.

commalg.ArithmeticFiberSupport

Exact or containing arithmetic support with bounded factorization evidence.

class ArithmeticFiberSupport

noether::commalg::ArithmeticFiberSupport · Direct Rust declaration

pub struct ArithmeticFiberSupport { /* 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/ArithmeticFiberSupport@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:12 · validation/unit/commalg/test_polynomial_ideal_facade.py:317 · validation/unit/commalg/test_polynomial_ideal_facade.py:323

commalg.ArithmeticFiberSupport.certificate

Return the certificate recorded by this ArithmeticFiberSupport.

Detailed contract

Multiplier identity when the result is a containing support.

(self) -> ArithmeticNullstellensatzCertificate | None

noether::commalg::ArithmeticFiberSupport::certificate · Direct Rust declaration

pub fn certificate(&self) -> Option<&ArithmeticNullstellensatzCertificate>
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/ArithmeticFiberSupport@1

Executable usage: examples/01_bezout_certificates.py:3 · examples/18_finite_field_schemes.py:14 · examples/18_finite_field_schemes.py:15 · validation/unit/commalg/test_acceptance_workload.py:135 · validation/unit/commalg/test_acceptance_workload.py:137 · validation/unit/commalg/test_acceptance_workload.py:138

commalg.ArithmeticFiberSupport.claim

Return the claim recorded by this ArithmeticFiberSupport.

Detailed contract

Mathematical strength of the support ideal.

(self) -> Literal['exact_contraction', 'certified_containment']

noether::commalg::ArithmeticFiberSupport::claim · Direct Rust declaration

pub fn claim(&self) -> ArithmeticFiberSupportClaim
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/ArithmeticFiberSupport@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:318 · validation/unit/commalg/test_polynomial_ideal_facade.py:331 · validation/bench/sage_workflows.py:5

commalg.ArithmeticFiberSupport.factorization

Return the factorization recorded by this ArithmeticFiberSupport.

Detailed contract

Bounded factorization evidence for the support generator.

(self) -> PartialIntegerFactorization

noether::commalg::ArithmeticFiberSupport::factorization · Direct Rust declaration

pub fn factorization(&self) -> &PartialIntegerFactorization
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/ArithmeticFiberSupport@1

Executable usage: examples/07_integer_factorization.py:10 · examples/07_integer_factorization.py:11 · examples/07_integer_factorization.py:12 · examples/07_integer_factorization.py:15 · examples/distributed/f2_linear_algebra.py:47 · examples/distributed/f2_linear_algebra.py:48

commalg.ArithmeticFiberSupport.from_json

Deserialize and revalidate a ArithmeticFiberSupport from typed JSON.

Detailed contract

Deserialize and verify support, witness, and bounded factorization.

(cls, value: str) -> ArithmeticFiberSupport

noether::commalg::ArithmeticFiberSupport::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/ArithmeticFiberSupport@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.ArithmeticFiberSupport.generators

Return the generators recorded by this ArithmeticFiberSupport.

Detailed contract

Generator family defining the fibers under study.

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

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

pub fn generators(&self) -> &[Polynomial<IntegerRing>]
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/ArithmeticFiberSupport@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.ArithmeticFiberSupport.is_exact

Test whether this ArithmeticFiberSupport is exact.

Detailed contract

Whether the support ideal is the exact contraction to Z.

(self) -> bool

noether::commalg::ArithmeticFiberSupport::is_exact · Direct Rust declaration

pub fn is_exact(&self) -> bool
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/ArithmeticFiberSupport@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:319 · validation/unit/commalg/test_polynomial_ideal_facade.py:332 · validation/unit/homalg/test_padic_homology_facade.py:36 · validation/unit/numthy/test_padic_ring_facade.py:49 · validation/unit/numthy/test_padic_ring_facade.py:52 · validation/unit/test_ball_facade.py:19

commalg.ArithmeticFiberSupport.prime_support_is_complete

Return the prime support is complete recorded by this ArithmeticFiberSupport.

Detailed contract

Whether every prime dividing the support generator has been identified.

(self) -> bool

noether::commalg::ArithmeticFiberSupport::prime_support_is_complete · Direct Rust declaration

pub fn prime_support_is_complete(&self) -> bool
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/ArithmeticFiberSupport@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:321

commalg.ArithmeticFiberSupport.proved_primes

Return the proved primes recorded by this ArithmeticFiberSupport.

Detailed contract

Proved prime factors currently known, including a proved-prime cofactor.

(self) -> list[tuple[int, int]]

noether::commalg::ArithmeticFiberSupport::proved_primes · Direct Rust declaration

pub fn proved_primes(&self) -> Vec<(Integer, u64)>
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/ArithmeticFiberSupport@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:320 · validation/unit/commalg/test_polynomial_ideal_facade.py:333

commalg.ArithmeticFiberSupport.ring

Return the ring recorded by this ArithmeticFiberSupport.

Detailed contract

Integral polynomial context whose fibers are being studied.

(self) -> PolynomialRing[IntegerRing]

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

pub fn ring(&self) -> &PolynomialRing<IntegerRing>
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/ArithmeticFiberSupport@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.ArithmeticFiberSupport.support_ideal

Return the support ideal recorded by this ArithmeticFiberSupport.

Detailed contract

Exact or containing principal support ideal, as specified by claim.

(self) -> Ideal[IntegerRing]

noether::commalg::ArithmeticFiberSupport::support_ideal · Direct Rust declaration

pub fn support_ideal(&self) -> &Ideal<IntegerRing>
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/ArithmeticFiberSupport@1

commalg.ArithmeticFiberSupport.to_json

Serialize this ArithmeticFiberSupport to its canonical typed JSON representation.

Detailed contract

Serialize the complete support evidence as typed JSON.

(self) -> str

noether::commalg::ArithmeticFiberSupport::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/ArithmeticFiberSupport@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.ArithmeticFiberSupport.unfactored_cofactor

Return the unfactored cofactor recorded by this ArithmeticFiberSupport.

Detailed contract

Positive cofactor whose prime divisors have not yet been identified.

(self) -> int | None

noether::commalg::ArithmeticFiberSupport::unfactored_cofactor · Direct Rust declaration

pub fn unfactored_cofactor(&self) -> Option<&Integer>
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/ArithmeticFiberSupport@1

commalg.ArithmeticFiberSupport.verify

Replay and verify the mathematical evidence carried by this ArithmeticFiberSupport.

Detailed contract

Verify support strength, multiplier evidence, and bounded factorization.

(self) -> None

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

pub fn verify(&self) -> Result<(), ArithmeticIdealError>
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/ArithmeticFiberSupport@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