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commalg · CurveFunctionField

17 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.CurveFunctionField

Canonical algebraic function field of a checked curve.

class CurveFunctionField

noether::commalg::CurveFunctionField · Direct Rust declaration

pub struct CurveFunctionField<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/CurveFunctionField@1

Executable usage: validation/unit/commalg/test_curve_facade.py:11 · validation/unit/commalg/test_curve_facade.py:30 · validation/unit/commalg/test_curve_facade.py:82

commalg.CurveFunctionField.algebraic_coordinate

Return the algebraic coordinate recorded by this CurveFunctionField.

Detailed contract

Coordinate algebraic over the rational subfield.

(self) -> int

noether::commalg::CurveFunctionField::algebraic_coordinate · Direct Rust declaration

pub fn algebraic_coordinate(&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/CurveFunctionField@1

commalg.CurveFunctionField.constant

Compute constant for this CurveFunctionField.

Detailed contract

Embed one constant-field element.

(self, value: CurveCoefficient) -> CurveFunction[C]

noether::commalg::CurveFunctionField::constant · Direct Rust declaration

pub fn constant( &self, value: R::Element, ) -> Result<CurveFunction<R>, CurveError>
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/CurveFunctionField@1

Executable usage: validation/unit/commalg/test_curve_facade.py:150 · validation/unit/commalg/test_normalization_facade.py:71 · validation/unit/commalg/test_polynomial_ideal_facade.py:304 · validation/unit/commalg/test_polynomial_ideal_facade.py:409 · validation/unit/commalg/test_polynomial_ideal_facade.py:413 · validation/unit/commalg/test_polynomial_ideal_facade.py:62

commalg.CurveFunctionField.coordinate

Compute coordinate for this CurveFunctionField.

Detailed contract

The ratio of a homogeneous coordinate to the selected patch coordinate.

(self, coordinate: int) -> CurveFunction[C]

noether::commalg::CurveFunctionField::coordinate · Direct Rust declaration

pub fn coordinate( &self, coordinate: usize, ) -> Result<CurveFunction<R>, CurveError>
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/CurveFunctionField@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:507 · validation/unit/numthy/test_qadic_field_facade.py:51 · validation/unit/numthy/test_qadic_ring_facade.py:68 · validation/parity/combinatorics/test_crystals_parity.py:169 · validation/bench/macaulay2_core.py:1544 · validation/bench/macaulay2_core.py:1725

commalg.CurveFunctionField.curve

Return the curve recorded by this CurveFunctionField.

Detailed contract

Source curve.

(self) -> Curve[C]

noether::commalg::CurveFunctionField::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/CurveFunctionField@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.CurveFunctionField.extension_degree

Return the extension degree recorded by this CurveFunctionField.

Detailed contract

Degree over the retained rational subfield.

(self) -> int

noether::commalg::CurveFunctionField::extension_degree · Direct Rust declaration

pub fn extension_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/CurveFunctionField@1

Executable usage: validation/unit/commalg/test_curve_facade.py:36

commalg.CurveFunctionField.from_homogeneous_polynomial

Construct a CurveFunctionField using homogeneous polynomial.

Detailed contract

Restrict one homogeneous-coordinate polynomial to the curve’s function field. Homogeneous coordinates are interpreted as ratios to the selected patch coordinate. For a homogeneous polynomial this differs from the original section by the corresponding patch-coordinate power, exactly as needed when forming a ratio of equal-degree sections.

(self, value: Polynomial[C]) -> CurveFunction[C]

noether::commalg::CurveFunctionField::from_homogeneous_polynomial · Direct Rust declaration

pub fn from_homogeneous_polynomial( &self, polynomial: &Polynomial<R>, ) -> Result<CurveFunction<R>, CurveError>
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/CurveFunctionField@1

Executable usage: validation/unit/commalg/test_curve_facade.py:53

commalg.CurveFunctionField.from_json

Deserialize and revalidate a CurveFunctionField from typed JSON.

Detailed contract

Restore the curve and deterministically rebuild its function field.

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

noether::commalg::CurveFunctionField::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>where R: DeserializeOwned, R::Element: DeserializeOwned,
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/CurveFunctionField@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.CurveFunctionField.homogeneous_fraction

Compute homogeneous fraction for this CurveFunctionField.

Detailed contract

Ratio of two equal-degree homogeneous sections restricted to the curve.

(self, numerator: Polynomial[C], denominator: Polynomial[C]) -> CurveFunction[C]

noether::commalg::CurveFunctionField::homogeneous_fraction · Direct Rust declaration

pub fn homogeneous_fraction( &self, numerator: &Polynomial<R>, denominator: &Polynomial<R>, ) -> Result<CurveFunction<R>, CurveError>
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/CurveFunctionField@1

Executable usage: validation/unit/commalg/test_curve_facade.py:52

commalg.CurveFunctionField.one

Compute one for this CurveFunctionField.

(self) -> CurveFunction[C]

noether::commalg::CurveFunctionField::one · Direct Rust declaration

fn one(&self) -> Self::Element
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/CurveFunctionField@1

Executable usage: examples/02_ball_arithmetic.py:13 · examples/11_qadic_witt_vectors.py:11 · examples/24_homology_and_persistence.py:3 · examples/31_local_fields_and_precision.py:27 · validation/unit/combinatorics/test_poset.py:3 · validation/unit/commalg/test_acceptance_workload.py:37

commalg.CurveFunctionField.patch_coordinate

Return the patch coordinate recorded by this CurveFunctionField.

Detailed contract

Homogeneous coordinate set to one in the selected affine chart.

(self) -> int

noether::commalg::CurveFunctionField::patch_coordinate · Direct Rust declaration

pub fn patch_coordinate(&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/CurveFunctionField@1

commalg.CurveFunctionField.to_json

Serialize this CurveFunctionField to its canonical typed JSON representation.

Detailed contract

Serialize the owning curve; the canonical field presentation is derived.

(self) -> str

noether::commalg::CurveFunctionField::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>where R: Serialize, R::Element: Serialize,
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/CurveFunctionField@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.CurveFunctionField.transcendental_coordinate

Return the transcendental coordinate recorded by this CurveFunctionField.

Detailed contract

Coordinate used as the transcendental generator.

(self) -> int

noether::commalg::CurveFunctionField::transcendental_coordinate · Direct Rust declaration

pub fn transcendental_coordinate(&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/CurveFunctionField@1

commalg.CurveFunctionField.verify

Replay and verify the mathematical evidence carried by this CurveFunctionField.

Detailed contract

Recheck the source curve, monic modulus, degree, and coordinate conversion.

(self) -> None

noether::commalg::CurveFunctionField::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/CurveFunctionField@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

commalg.CurveFunctionField.x

Compute x for this CurveFunctionField.

Detailed contract

The affine coordinate x in the chosen chart.

(self) -> CurveFunction[C]

noether::commalg::CurveFunctionField::x · Direct Rust declaration

pub fn x(&self) -> Result<CurveFunction<R>, CurveError>
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/CurveFunctionField@1

Executable usage: examples/10_padic_hensel_lifting.py:11 · examples/10_padic_hensel_lifting.py:12 · examples/10_padic_hensel_lifting.py:13 · examples/12_number_field_ideals.py:11 · examples/12_number_field_ideals.py:12 · examples/14_weighted_polynomials.py:10

commalg.CurveFunctionField.y

Compute y for this CurveFunctionField.

Detailed contract

The affine coordinate y in the chosen chart.

(self) -> CurveFunction[C]

noether::commalg::CurveFunctionField::y · Direct Rust declaration

pub fn y(&self) -> Result<CurveFunction<R>, CurveError>
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/CurveFunctionField@1

Executable usage: examples/14_weighted_polynomials.py:10 · examples/14_weighted_polynomials.py:11 · examples/14_weighted_polynomials.py:14 · examples/14_weighted_polynomials.py:17 · examples/15_groebner_twisted_cubic.py:10 · examples/15_groebner_twisted_cubic.py:11

commalg.CurveFunctionField.zero

Compute zero for this CurveFunctionField.

(self) -> CurveFunction[C]

noether::commalg::CurveFunctionField::zero · Direct Rust declaration

fn zero(&self) -> Self::Element
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/CurveFunctionField@1

Executable usage: examples/08_finite_field_extensions.py:11 · examples/10_padic_hensel_lifting.py:15 · examples/40_exterior_algebra.py:26 · examples/distributed/f2_linear_algebra.py:28 · examples/distributed/f2_linear_algebra.py:29 · examples/distributed/f2_linear_algebra.py:30