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commalg · SmoothProjectiveSurfaceCohomology

12 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.SmoothProjectiveSurfaceCohomology

Cohomological profile that establishes the degree needed for Frobenius.

class SmoothProjectiveSurfaceCohomology

noether::commalg::SmoothProjectiveSurfaceCohomology · Direct Rust declaration

pub struct SmoothProjectiveSurfaceCohomology { /* 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/SmoothProjectiveSurfaceCohomology@1

commalg.SmoothProjectiveSurfaceCohomology.branch_degree

Return the branch degree recorded by this SmoothProjectiveSurfaceCohomology.

Detailed contract

Degree of the smooth plane branch curve.

(self) -> int

noether::commalg::SmoothProjectiveSurfaceCohomology::branch_degree · Direct Rust declaration

pub fn branch_degree(&self) -> 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/SmoothProjectiveSurfaceCohomology@1

commalg.SmoothProjectiveSurfaceCohomology.branch_genus

Return the branch genus recorded by this SmoothProjectiveSurfaceCohomology.

Detailed contract

Genus (d-1)(d-2)/2 of the smooth plane branch curve.

(self) -> int

noether::commalg::SmoothProjectiveSurfaceCohomology::branch_genus · Direct Rust declaration

pub fn branch_genus(&self) -> &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/SmoothProjectiveSurfaceCohomology@1

commalg.SmoothProjectiveSurfaceCohomology.cover

Return the cover recorded by this SmoothProjectiveSurfaceCohomology.

Detailed contract

Cyclic-cover presentation establishing this profile.

(self) -> CyclicHypersurfaceCover[FiniteField]

noether::commalg::SmoothProjectiveSurfaceCohomology::cover · Direct Rust declaration

pub fn cover(&self) -> &CyclicHypersurfaceCover<FiniteField>
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/SmoothProjectiveSurfaceCohomology@1

Executable usage: validation/unit/commalg/test_surface_facade.py:38 · validation/unit/commalg/test_surface_facade.py:40 · validation/unit/commalg/test_surface_facade.py:43 · validation/unit/commalg/test_surface_facade.py:45 · validation/unit/commalg/test_surface_facade.py:61 · validation/unit/commalg/test_surface_facade.py:71

commalg.SmoothProjectiveSurfaceCohomology.euler_characteristic

Return the euler characteristic recorded by this SmoothProjectiveSurfaceCohomology.

Detailed contract

Euler characteristic of the covering surface.

(self) -> int

noether::commalg::SmoothProjectiveSurfaceCohomology::euler_characteristic · Direct Rust declaration

pub fn euler_characteristic(&self) -> &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/SmoothProjectiveSurfaceCohomology@1

Executable usage: validation/unit/combinatorics/test_poset.py:143 · validation/unit/topology/test_simplicial_complex_facade.py:62 · validation/unit/topology/test_simplicial_set_facade.py:262 · validation/unit/topology/test_simplicial_set_facade.py:53 · validation/corpus/topology/bowtie-complex-homology.py:15 · validation/corpus/topology/bowtie-complex-homology.py:25

commalg.SmoothProjectiveSurfaceCohomology.frobenius_polynomial

Compute frobenius polynomial for this SmoothProjectiveSurfaceCohomology.

Detailed contract

Recover det(T-F | H^2) from sufficient consecutive exact counts.

(self, counts: PointCountSequence) -> FrobeniusPolynomial

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

pub fn frobenius_polynomial( &self, counts: &PointCountSequence, ) -> Result<FrobeniusPolynomial, 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/SmoothProjectiveSurfaceCohomology@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.SmoothProjectiveSurfaceCohomology.from_json

Deserialize and revalidate a SmoothProjectiveSurfaceCohomology from typed JSON.

Detailed contract

Restore and recompute the complete geometric evidence.

(cls, value: str) -> SmoothProjectiveSurfaceCohomology

noether::commalg::SmoothProjectiveSurfaceCohomology::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/SmoothProjectiveSurfaceCohomology@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.SmoothProjectiveSurfaceCohomology.hyperplane_frobenius_eigenvalue

Return the hyperplane frobenius eigenvalue recorded by this SmoothProjectiveSurfaceCohomology.

Detailed contract

Frobenius eigenvalue supplied by the pullback of a base hyperplane.

(self) -> int

noether::commalg::SmoothProjectiveSurfaceCohomology::hyperplane_frobenius_eigenvalue · Direct Rust declaration

pub fn hyperplane_frobenius_eigenvalue(&self) -> 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/SmoothProjectiveSurfaceCohomology@1

commalg.SmoothProjectiveSurfaceCohomology.required_point_count_degree

Return the required point count degree recorded by this SmoothProjectiveSurfaceCohomology.

Detailed contract

Number of consecutive counts normally sufficient for reconstruction. The pullback of a line is defined over the base field and contributes the known Frobenius eigenvalue q. Poincare duality then reconstructs the remaining coefficients from ceil(b_2/2) traces. In a degenerate zero-coefficient case, one further count may be requested to distinguish the two possible functional-equation signs.

(self) -> int

noether::commalg::SmoothProjectiveSurfaceCohomology::required_point_count_degree · Direct Rust declaration

pub fn required_point_count_degree(&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/SmoothProjectiveSurfaceCohomology@1

Executable usage: validation/unit/commalg/test_surface_facade.py:63

commalg.SmoothProjectiveSurfaceCohomology.second_betti_number

Return the second betti number recorded by this SmoothProjectiveSurfaceCohomology.

Detailed contract

Dimension of second etale cohomology.

(self) -> int

noether::commalg::SmoothProjectiveSurfaceCohomology::second_betti_number · Direct Rust declaration

pub fn second_betti_number(&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/SmoothProjectiveSurfaceCohomology@1

Executable usage: validation/unit/commalg/test_surface_facade.py:62

commalg.SmoothProjectiveSurfaceCohomology.to_json

Serialize this SmoothProjectiveSurfaceCohomology to its canonical typed JSON representation.

Detailed contract

Serialize the complete geometric evidence.

(self) -> str

noether::commalg::SmoothProjectiveSurfaceCohomology::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/SmoothProjectiveSurfaceCohomology@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.SmoothProjectiveSurfaceCohomology.verify

Replay and verify the mathematical evidence carried by this SmoothProjectiveSurfaceCohomology.

Detailed contract

Recompute the smoothness and cohomological formula.

(self) -> None

noether::commalg::SmoothProjectiveSurfaceCohomology::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/SmoothProjectiveSurfaceCohomology@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