Skip to content

commalg · CurveClosedPoint

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.CurveClosedPoint

A closed point (place) on one curve.

class CurveClosedPoint

noether::commalg::CurveClosedPoint · Direct Rust declaration

pub struct CurveClosedPoint<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/CurveClosedPoint@1

commalg.CurveClosedPoint.curve

Return the curve recorded by this CurveClosedPoint.

Detailed contract

Owning curve.

(self) -> Curve[C]

noether::commalg::CurveClosedPoint::curve · Direct Rust declaration

pub fn curve(&self) -> &Curve<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/CurveClosedPoint@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.CurveClosedPoint.defining_ideal

Return the defining ideal recorded by this CurveClosedPoint.

(self) -> Ideal[PolynomialRing[C]]

noether::commalg::CurveClosedPoint · 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/CurveClosedPoint@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:107 · validation/unit/commalg/test_acceptance_workload.py:114 · validation/unit/commalg/test_affine_normalization_facade.py:108 · validation/unit/commalg/test_affine_normalization_facade.py:163 · validation/unit/commalg/test_affine_normalization_facade.py:182 · validation/unit/commalg/test_affine_normalization_facade.py:89

commalg.CurveClosedPoint.degree

Return the degree recorded by this CurveClosedPoint.

Detailed contract

Residue degree over the constant field.

(self) -> int

noether::commalg::CurveClosedPoint::degree · Direct Rust declaration

pub fn degree(&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/CurveClosedPoint@1

Executable usage: examples/14_weighted_polynomials.py:16 · examples/14_weighted_polynomials.py:18 · examples/15_groebner_twisted_cubic.py:15 · examples/15_groebner_twisted_cubic.py:28 · examples/31_local_fields_and_precision.py:37 · examples/33_simplicial_cohomology_operations.py:26

commalg.CurveClosedPoint.from_json

Deserialize and revalidate a CurveClosedPoint from typed JSON.

Detailed contract

Restore and prove maximality in a deterministic affine chart. Restore and prove a degree-one place from its rational coordinates.

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

noether::commalg::CurveClosedPoint::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/CurveClosedPoint@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.CurveClosedPoint.rational_point

Return the rational point recorded by this CurveClosedPoint.

Detailed contract

Retained rational coordinates, when this point was embedded from them.

(self) -> CurvePoint[C] | None

noether::commalg::CurveClosedPoint::rational_point · Direct Rust declaration

pub fn rational_point(&self) -> Option<&CurvePoint<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/CurveClosedPoint@1

Executable usage: validation/unit/numthy/test_elliptic_curve_facade.py:20 · validation/unit/numthy/test_elliptic_curve_facade.py:21 · validation/unit/numthy/test_elliptic_curve_facade.py:22 · validation/unit/numthy/test_elliptic_curve_facade.py:23 · validation/unit/numthy/test_elliptic_curve_facade.py:41

commalg.CurveClosedPoint.to_json

Serialize this CurveClosedPoint to its canonical typed JSON representation.

Detailed contract

Serialize this place by its checked curve and homogeneous prime ideal. Serialize a degree-one place over this exact function field.

(self) -> str

noether::commalg::CurveClosedPoint::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/CurveClosedPoint@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.CurveClosedPoint.verify

Replay and verify the mathematical evidence carried by this CurveClosedPoint.

Detailed contract

Recheck incidence, dimension, degree, and any rational coordinates.

(self) -> None

noether::commalg::CurveClosedPoint::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/CurveClosedPoint@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