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commalg · RiemannRochSpace

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

Worked example

The complete checked program Reproduce Fisher's order-five elliptic-function normalization uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Reproduce Fisher's order-five elliptic-function normalization."""

from noether import Q
from noether.commalg import RationalFunctionField, WeilDivisor
from noether.numthy import EllipticCurve, NthPowerClassGroup


def main() -> None:
    coefficients = RationalFunctionField(Q, "lambda")
    parameter = coefficients.generator()
    curve = EllipticCurve(
        coefficients,
        [1 - parameter, -parameter, -parameter, 0, 0],
    )
    point = curve.point(0, 0)
    point.verify_exact_order(5)

    miller = curve.miller_function(point, 5)
    divisor = WeilDivisor(
        curve.underlying_curve,
        [
            (curve.origin().underlying_curve_point.as_closed_point(), 5),
            (point.underlying_curve_point.as_closed_point(), -5),
        ],
    )
    function = divisor.riemann_roch_space().basis[0]
    assert (function / miller.function).as_constant() is not None

    evaluation_at_two = function.evaluate_at((2 * point).underlying_curve_point)
    evaluation_at_four = function.evaluate_at((4 * point).underlying_curve_point)
    assert evaluation_at_two.value is not None
    assert evaluation_at_four.value is not None
    classes = NthPowerClassGroup(coefficients, 5)
    normalization = classes.class_of(evaluation_at_two.value) ** -2 * classes.class_of(
        evaluation_at_four.value
    )
    assert normalization.is_identity()
    print("order 5 verified; Riemann--Roch and Miller agree; normalization is trivial")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/26_elliptic_descent_functions.py.

commalg.RiemannRochSpace

A canonical verified Riemann--Roch space.

class RiemannRochSpace

noether::commalg::RiemannRochSpace · Direct Rust declaration

pub struct RiemannRochSpace<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/RiemannRochSpace@1

Executable usage: validation/unit/commalg/test_curve_facade.py:126 · validation/unit/commalg/test_curve_facade.py:15 · validation/unit/commalg/test_curve_facade.py:74 · validation/unit/commalg/test_curve_facade.py:87

commalg.RiemannRochSpace.basis

Return the basis recorded by this RiemannRochSpace.

Detailed contract

Deterministically ordered basis in the curve function field.

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

noether::commalg::RiemannRochSpace::basis · Direct Rust declaration

pub fn basis(&self) -> &[CurveFunction<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/RiemannRochSpace@1

Executable usage: examples/26_elliptic_descent_functions.py:28 · examples/30_modules_and_resolutions.py:27 · examples/30_modules_and_resolutions.py:28 · examples/30_modules_and_resolutions.py:29 · examples/33_simplicial_cohomology_operations.py:26 · examples/33_simplicial_cohomology_operations.py:27

commalg.RiemannRochSpace.curve

Return the curve recorded by this RiemannRochSpace.

(self) -> Curve[C]

noether::commalg::RiemannRochSpace · Python projection

Python projection convenience over this Rust mathematical type
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/RiemannRochSpace@1

Executable usage: examples/26_elliptic_descent_functions.py:13 · examples/26_elliptic_descent_functions.py:17 · examples/26_elliptic_descent_functions.py:20 · examples/26_elliptic_descent_functions.py:22 · examples/26_elliptic_descent_functions.py:24 · examples/37_complete_intersection_theory.py:27

commalg.RiemannRochSpace.dimension

Return the dimension recorded by this RiemannRochSpace.

Detailed contract

Dimension over the constant field.

(self) -> int

noether::commalg::RiemannRochSpace::dimension · Direct Rust declaration

pub fn dimension(&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/RiemannRochSpace@1

Executable usage: examples/03_exact_linear_systems.py:15 · examples/03_exact_linear_systems.py:16 · examples/15_groebner_twisted_cubic.py:14 · examples/15_groebner_twisted_cubic.py:28 · examples/27_weight_characters.py:25 · examples/34_finite_group_cohomology.py:13

commalg.RiemannRochSpace.divisor

Return the divisor recorded by this RiemannRochSpace.

Detailed contract

Source divisor.

(self) -> WeilDivisor[C]

noether::commalg::RiemannRochSpace::divisor · Direct Rust declaration

pub fn divisor(&self) -> &WeilDivisor<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/RiemannRochSpace@1

Executable usage: examples/26_elliptic_descent_functions.py:21 · examples/26_elliptic_descent_functions.py:28 · validation/unit/commalg/test_curve_facade.py:114 · validation/unit/commalg/test_curve_facade.py:116 · validation/unit/commalg/test_curve_facade.py:122 · validation/unit/commalg/test_curve_facade.py:125

commalg.RiemannRochSpace.from_json

Deserialize and revalidate a RiemannRochSpace from typed JSON.

Detailed contract

Restore the divisor and recompute the deterministic section basis.

(cls, value: str) -> RiemannRochSpace[CurveBase]

noether::commalg::RiemannRochSpace::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/RiemannRochSpace@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.RiemannRochSpace.to_json

Serialize this RiemannRochSpace to its canonical typed JSON representation.

Detailed contract

Serialize the source divisor; the canonical basis is derived evidence.

(self) -> str

noether::commalg::RiemannRochSpace::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/RiemannRochSpace@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.RiemannRochSpace.verify

Replay and verify the mathematical evidence carried by this RiemannRochSpace.

Detailed contract

Recompute the complete ideal-theoretic section calculation.

(self) -> None

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

pub fn verify(&self) -> Result<(), VerificationError>
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/RiemannRochSpace@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