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commalg · GeometricDivisorRankUpperBound

9 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.GeometricDivisorRankUpperBound

Tate-style upper bound on the geometric divisor rank.

class GeometricDivisorRankUpperBound

noether::commalg::GeometricDivisorRankUpperBound · Direct Rust declaration

pub struct GeometricDivisorRankUpperBound { /* 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/GeometricDivisorRankUpperBound@1

commalg.GeometricDivisorRankUpperBound.certify_exact

Compute certify exact for this GeometricDivisorRankUpperBound.

Detailed contract

Combine with independently verified divisor classes when the bounds match.

(self, lower: DivisorRankLowerBound) -> GeometricDivisorRank

noether::commalg::GeometricDivisorRankUpperBound::certify_exact · Direct Rust declaration

pub fn certify_exact( &self, lower: &DivisorRankLowerBound, ) -> Result<GeometricDivisorRank, FrobeniusError>
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/GeometricDivisorRankUpperBound@1

Executable usage: validation/unit/commalg/test_surface_facade.py:76

commalg.GeometricDivisorRankUpperBound.cyclotomic_factors

Return the cyclotomic factors recorded by this GeometricDivisorRankUpperBound.

Detailed contract

Complete cyclotomic factors and multiplicities.

(self) -> list[TateCyclotomicFactor]

noether::commalg::GeometricDivisorRankUpperBound::cyclotomic_factors · Direct Rust declaration

pub fn cyclotomic_factors(&self) -> &[TateCyclotomicFactor]
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/GeometricDivisorRankUpperBound@1

Executable usage: validation/bench/surface_frobenius.py:132

commalg.GeometricDivisorRankUpperBound.frobenius_polynomial

Return the frobenius polynomial recorded by this GeometricDivisorRankUpperBound.

Detailed contract

Frobenius polynomial from which the bound was obtained.

(self) -> FrobeniusPolynomial

noether::commalg::GeometricDivisorRankUpperBound::frobenius_polynomial · Direct Rust declaration

pub fn frobenius_polynomial(&self) -> &FrobeniusPolynomial
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/GeometricDivisorRankUpperBound@1

Executable usage: validation/unit/commalg/test_surface_facade.py:64 · validation/parity/commalg/test_surface_magma_parity.py:66 · validation/bench/surface_frobenius.py:91

commalg.GeometricDivisorRankUpperBound.from_json

Deserialize and revalidate a GeometricDivisorRankUpperBound from typed JSON.

Detailed contract

Restore and exactly reverify the upper bound.

(cls, value: str) -> GeometricDivisorRankUpperBound

noether::commalg::GeometricDivisorRankUpperBound::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/GeometricDivisorRankUpperBound@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.GeometricDivisorRankUpperBound.tate_normalized_polynomial

Return the tate normalized polynomial recorded by this GeometricDivisorRankUpperBound.

Detailed contract

Monic polynomial whose roots are the Frobenius eigenvalues divided by q.

(self) -> Polynomial[RationalField]

noether::commalg::GeometricDivisorRankUpperBound::tate_normalized_polynomial · Direct Rust declaration

pub fn tate_normalized_polynomial(&self) -> &Polynomial<RationalField>
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/GeometricDivisorRankUpperBound@1

commalg.GeometricDivisorRankUpperBound.to_json

Serialize this GeometricDivisorRankUpperBound to its canonical typed JSON representation.

Detailed contract

Serialize the normalized factorization evidence.

(self) -> str

noether::commalg::GeometricDivisorRankUpperBound::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/GeometricDivisorRankUpperBound@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.GeometricDivisorRankUpperBound.upper_bound

Return the upper bound recorded by this GeometricDivisorRankUpperBound.

Detailed contract

Certified upper bound on geometric divisor rank.

(self) -> int

noether::commalg::GeometricDivisorRankUpperBound::upper_bound · Direct Rust declaration

pub fn upper_bound(&self) -> usize
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/GeometricDivisorRankUpperBound@1

Executable usage: validation/unit/commalg/test_surface_facade.py:75 · validation/bench/surface_frobenius.py:120 · validation/bench/surface_frobenius.py:132 · validation/bench/surface_frobenius.py:134 · validation/bench/surface_frobenius.py:89 · validation/bench/surface_frobenius.py:92

commalg.GeometricDivisorRankUpperBound.verify

Replay and verify the mathematical evidence carried by this GeometricDivisorRankUpperBound.

Detailed contract

Recompute normalization, factorization, and the complete cyclotomic search.

(self) -> None

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

pub fn verify(&self) -> Result<(), FrobeniusError>
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/GeometricDivisorRankUpperBound@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