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commalg · ZetaFunction

7 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.ZetaFunction

Exact rational zeta function of a supported smooth projective surface.

class ZetaFunction

noether::commalg::ZetaFunction · Direct Rust declaration

pub struct ZetaFunction { /* 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/ZetaFunction@1

Executable usage: validation/unit/commalg/test_surface_facade.py:18 · validation/unit/commalg/test_surface_facade.py:67

commalg.ZetaFunction.denominator

Return the denominator recorded by this ZetaFunction.

Detailed contract

Exact denominator in the canonical variable T.

(self) -> Polynomial[IntegerRing]

noether::commalg::ZetaFunction::denominator · Direct Rust declaration

pub fn denominator(&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/ZetaFunction@1

Executable usage: validation/unit/commalg/test_curve_facade.py:51 · validation/unit/commalg/test_curve_facade.py:52 · validation/unit/commalg/test_curve_facade.py:93 · validation/unit/commalg/test_curve_facade.py:96 · validation/unit/commalg/test_local_algebra.py:30 · validation/unit/commalg/test_local_algebra.py:46

commalg.ZetaFunction.frobenius_polynomial

Return the frobenius polynomial recorded by this ZetaFunction.

Detailed contract

Frobenius polynomial supplying the nontrivial Euler factor.

(self) -> FrobeniusPolynomial

noether::commalg::ZetaFunction::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/ZetaFunction@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.ZetaFunction.from_json

Deserialize and revalidate a ZetaFunction from typed JSON.

Detailed contract

Restore and exactly reverify a zeta function.

(cls, value: str) -> ZetaFunction

noether::commalg::ZetaFunction::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/ZetaFunction@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.ZetaFunction.numerator

Return the numerator recorded by this ZetaFunction.

Detailed contract

Exact numerator in the canonical variable T.

(self) -> Polynomial[IntegerRing]

noether::commalg::ZetaFunction::numerator · Direct Rust declaration

pub fn numerator(&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/ZetaFunction@1

Executable usage: validation/unit/commalg/test_curve_facade.py:51 · validation/unit/commalg/test_curve_facade.py:52 · validation/unit/commalg/test_curve_facade.py:93 · validation/unit/commalg/test_curve_facade.py:95 · validation/unit/commalg/test_local_algebra.py:29 · validation/unit/commalg/test_monomial_ideal_facade.py:86

commalg.ZetaFunction.to_json

Serialize this ZetaFunction to its canonical typed JSON representation.

Detailed contract

Serialize all Euler factors and their evidence.

(self) -> str

noether::commalg::ZetaFunction::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/ZetaFunction@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.ZetaFunction.verify

Replay and verify the mathematical evidence carried by this ZetaFunction.

Detailed contract

Rebuild the Euler factors and recheck every recorded point count.

(self) -> None

noether::commalg::ZetaFunction::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/ZetaFunction@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