commalg · CurveFunction¶

24 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.CurveFunction`¶

One canonical rational function on a checked curve.

PythonRust

class CurveFunction

noether::commalg::CurveFunction · Direct Rust declaration

pub struct CurveFunction<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/CurveFunction@1

Executable usage: validation/unit/commalg/test_curve_facade.py:31 · validation/unit/commalg/test_curve_facade.py:85 · validation/unit/commalg/test_curve_facade.py:9 · validation/unit/commalg/test_curve_facade.py:98

`commalg.CurveFunction.add`¶

Add another compatible value to this value.

Detailed contract

Returns the argument unchanged.

PythonRust

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

noether::commalg::CurveFunction::add · Direct Rust declaration

fn add(self, rhs: &CurveFunction<R>) -> Self::Output

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/CurveFunction@1

Executable usage: validation/unit/infra/test_discovery.py:211 · validation/unit/infra/test_public_api_index.py:293

`commalg.CurveFunction.mul`¶

Multiply this value by another compatible value.

PythonRust

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

noether::commalg::CurveFunction::mul · Direct Rust declaration

fn mul(self, rhs: &CurveFunction<R>) -> Self::Output

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/CurveFunction@1

Executable usage: validation/unit/infra/test_public_api_index.py:214

`commalg.CurveFunction.neg`¶

Return the additive inverse of this value.

PythonRust

(self) -> CurveFunction[C]

noether::commalg::CurveFunction::neg · Direct Rust declaration

fn neg(self) -> Self::Output

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/CurveFunction@1

`commalg.CurveFunction.pow`¶

Raise this value to the supplied power.

PythonRust

(self, exponent: int, modulus: None=None) -> CurveFunction[C]

noether::commalg::CurveFunction · 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/CurveFunction@1

`commalg.CurveFunction.radd`¶

Add this value to a compatible value on its left.

PythonRust

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

noether::commalg::CurveFunction · 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/CurveFunction@1

`commalg.CurveFunction.rmul`¶

Multiply this value by a compatible value on its left.

PythonRust

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

noether::commalg::CurveFunction · 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/CurveFunction@1

`commalg.CurveFunction.rsub`¶

Subtract this value from a compatible value on its left.

PythonRust

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

noether::commalg::CurveFunction · 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/CurveFunction@1

`commalg.CurveFunction.rtruediv`¶

Divide a compatible value on the left by this value.

PythonRust

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

noether::commalg::CurveFunction · 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/CurveFunction@1

`commalg.CurveFunction.sub`¶

Subtract another compatible value from this value.

PythonRust

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

noether::commalg::CurveFunction::sub · Direct Rust declaration

fn sub(self, rhs: &CurveFunction<R>) -> Self::Output

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/CurveFunction@1

Executable usage: validation/unit/infra/test_discovery.py:212

`commalg.CurveFunction.truediv`¶

Divide this value by another compatible value.

PythonRust

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

noether::commalg::CurveFunction · 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/CurveFunction@1

`commalg.CurveFunction.as_constant`¶

View this CurveFunction as constant.

Detailed contract

Recover the constant-field value when this function is constant.

PythonRust

(self) -> CurveCoefficient | None

noether::commalg::CurveFunction::as_constant · Direct Rust declaration

pub fn as_constant(&self) -> Result<Option<R::Element>, 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/CurveFunction@1

Executable usage: examples/26_elliptic_descent_functions.py:29 · validation/unit/commalg/test_curve_facade.py:124 · validation/unit/numthy/test_elliptic_curve_facade.py:82

`commalg.CurveFunction.evaluate_at`¶

Compute evaluate at for this CurveFunction.

Detailed contract

Evaluate at a rational point with distinct regular, zero, and pole results.

PythonRust

(self, point: CurvePoint[C], *, limit: int=256) -> CurveFunctionEvaluation

noether::commalg::CurveFunction::evaluate_at · Direct Rust declaration

pub fn evaluate_at( &self, point: &CurvePoint<R>, ) -> Result<CurveFunctionEvaluation<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/CurveFunction@1

Executable usage: examples/26_elliptic_descent_functions.py:31 · examples/26_elliptic_descent_functions.py:32 · validation/unit/commalg/test_curve_facade.py:137 · validation/unit/commalg/test_curve_facade.py:39 · validation/unit/commalg/test_curve_facade.py:40 · validation/unit/commalg/test_curve_facade.py:41

`commalg.CurveFunction.field`¶

Return the field recorded by this CurveFunction.

Detailed contract

Owning curve function field.

PythonRust

(self) -> CurveFunctionField[C]

noether::commalg::CurveFunction::field · Direct Rust declaration

pub fn field(&self) -> &CurveFunctionField<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/CurveFunction@1

Executable usage: examples/08_finite_field_extensions.py:10 · examples/08_finite_field_extensions.py:11 · examples/08_finite_field_extensions.py:12 · examples/08_finite_field_extensions.py:13 · examples/08_finite_field_extensions.py:14 · examples/08_finite_field_extensions.py:3

`commalg.CurveFunction.from_json`¶

Deserialize and revalidate a CurveFunction from typed JSON.

Detailed contract

Restore, reduce, and reject a noncanonical homogeneous presentation.

PythonRust

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

noether::commalg::CurveFunction::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/CurveFunction@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.CurveFunction.homogeneous_representation`¶

Compute homogeneous representation for this CurveFunction.

Detailed contract

Canonical equal-degree homogeneous numerator and denominator sections. The zero function is represented by (0,1). Every nonzero function is first reduced in the admitted K(x)[y] presentation and then cleared of coefficient denominators deterministically.

PythonRust

(self) -> tuple[Polynomial[C], Polynomial[C]]

noether::commalg::CurveFunction::homogeneous_representation · Direct Rust declaration

pub fn homogeneous_representation( &self, ) -> Result<(Polynomial<R>, Polynomial<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/CurveFunction@1

Executable usage: validation/unit/commalg/test_curve_facade.py:51 · validation/unit/commalg/test_curve_facade.py:93

`commalg.CurveFunction.inverse`¶

Compute inverse for this CurveFunction.

PythonRust

(self) -> CurveFunction[C]

noether::commalg::CurveFunction · 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/CurveFunction@1

Executable usage: examples/distributed/f2_linear_algebra.py:37 · examples/distributed/f2_linear_algebra.py:39 · validation/unit/commalg/test_context_tower_workload.py:121 · validation/unit/commalg/test_context_tower_workload.py:122 · validation/unit/commalg/test_context_tower_workload.py:75 · validation/unit/commalg/test_context_tower_workload.py:77

`commalg.CurveFunction.is_one`¶

Test whether this CurveFunction is one.

Detailed contract

Whether this function is one.

PythonRust

(self) -> bool

noether::commalg::CurveFunction::is_one · Direct Rust declaration

pub fn is_one(&self) -> bool

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/CurveFunction@1

Executable usage: validation/unit/commalg/test_rational_function_facade.py:31 · validation/unit/commalg/test_residue_ring_facade.py:25 · validation/unit/numthy/test_number_field_facade.py:29 · validation/corpus/commalg/evolution-algebra-isomorphism.py:34 · validation/corpus/numthy/diagonal-quadric-local-density-sums.py:36

`commalg.CurveFunction.is_zero`¶

Test whether this CurveFunction is zero.

Detailed contract

Whether this function is zero.

PythonRust

(self) -> bool

noether::commalg::CurveFunction::is_zero · Direct Rust declaration

pub fn is_zero(&self) -> bool

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/CurveFunction@1

Executable usage: examples/17_weighted_power_series.py:16 · examples/17_weighted_power_series.py:17 · examples/39_kahler_differentials.py:20 · validation/unit/commalg/test_graded_commutative_facade.py:26 · validation/unit/commalg/test_kahler_differentials.py:113 · validation/unit/commalg/test_kahler_differentials.py:38

`commalg.CurveFunction.order_at`¶

Compute order at for this CurveFunction.

Detailed contract

Exact discrete-valuation order at a checked rational point. The local implicit-function expansion is deliberately bounded. Use order_at_with_limit when a larger order is expected.

PythonRust

(self, point: CurvePoint[C], *, limit: int=256) -> CurveFunctionOrder

noether::commalg::CurveFunction::order_at · Direct Rust declaration

pub fn order_at( &self, point: &CurvePoint<R>, ) -> Result<CurveFunctionOrder, 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/CurveFunction@1

Executable usage: validation/unit/commalg/test_curve_facade.py:47

`commalg.CurveFunction.principal_divisor`¶

Compute principal divisor for this CurveFunction.

Detailed contract

Complete principal Weil divisor over Q, including non-rational closed points. Complete principal Weil divisor over a finite field, with residue-field places retained.

PythonRust

(self) -> WeilDivisor[C]

noether::commalg::CurveFunction::principal_divisor · Direct Rust declaration

pub fn principal_divisor( &self, ) -> Result<WeilDivisor<RationalField>, CurveError>
pub fn principal_divisor(&self) -> Result<WeilDivisor<FiniteField>, 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/CurveFunction@1

Executable usage: validation/unit/commalg/test_curve_facade.py:135 · validation/unit/commalg/test_curve_facade.py:55

`commalg.CurveFunction.to_json`¶

Serialize this CurveFunction to its canonical typed JSON representation.

Detailed contract

Serialize a canonical equal-degree homogeneous fraction on its curve.

PythonRust

(self) -> str

noether::commalg::CurveFunction::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/CurveFunction@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.CurveFunction.verify`¶

Replay and verify the mathematical evidence carried by this CurveFunction.

Detailed contract

Recompute reduction modulo the defining equation.

PythonRust

(self) -> None

noether::commalg::CurveFunction::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/CurveFunction@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.CurveFunction.verify_principal_divisor`¶

Compute verify principal divisor for this CurveFunction.

Detailed contract

Independently recompute and compare a proposed principal divisor.

PythonRust

(self, divisor: WeilDivisor[C]) -> None

noether::commalg::CurveFunction::verify_principal_divisor · Direct Rust declaration

pub fn verify_principal_divisor( &self, divisor: &WeilDivisor<RationalField>, ) -> Result<(), VerificationError>
pub fn verify_principal_divisor( &self, divisor: &WeilDivisor<FiniteField>, ) -> 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/CurveFunction@1

Executable usage: validation/unit/commalg/test_curve_facade.py:141 · validation/unit/commalg/test_curve_facade.py:63

commalg · CurveFunction¶

Worked example¶

commalg.CurveFunction¶

commalg.CurveFunction.__add__¶

commalg.CurveFunction.__mul__¶

commalg.CurveFunction.__neg__¶

commalg.CurveFunction.__pow__¶

commalg.CurveFunction.__radd__¶

commalg.CurveFunction.__rmul__¶

commalg.CurveFunction.__rsub__¶

commalg.CurveFunction.__rtruediv__¶

commalg.CurveFunction.__sub__¶

commalg.CurveFunction.__truediv__¶

commalg.CurveFunction.as_constant¶

commalg.CurveFunction.evaluate_at¶

commalg.CurveFunction.field¶

commalg.CurveFunction.from_json¶

commalg.CurveFunction.homogeneous_representation¶

commalg.CurveFunction.inverse¶

commalg.CurveFunction.is_one¶

commalg.CurveFunction.is_zero¶

commalg.CurveFunction.order_at¶

commalg.CurveFunction.principal_divisor¶

commalg.CurveFunction.to_json¶

commalg.CurveFunction.verify¶

commalg.CurveFunction.verify_principal_divisor¶

`commalg.CurveFunction`¶

`commalg.CurveFunction.add`¶

`commalg.CurveFunction.mul`¶

`commalg.CurveFunction.neg`¶

`commalg.CurveFunction.pow`¶

`commalg.CurveFunction.radd`¶

`commalg.CurveFunction.rmul`¶

`commalg.CurveFunction.rsub`¶

`commalg.CurveFunction.rtruediv`¶

`commalg.CurveFunction.sub`¶

`commalg.CurveFunction.truediv`¶

`commalg.CurveFunction.as_constant`¶

`commalg.CurveFunction.evaluate_at`¶

`commalg.CurveFunction.field`¶

`commalg.CurveFunction.from_json`¶

`commalg.CurveFunction.homogeneous_representation`¶

`commalg.CurveFunction.inverse`¶

`commalg.CurveFunction.is_one`¶

`commalg.CurveFunction.is_zero`¶

`commalg.CurveFunction.order_at`¶

`commalg.CurveFunction.principal_divisor`¶

`commalg.CurveFunction.to_json`¶

`commalg.CurveFunction.verify`¶

`commalg.CurveFunction.verify_principal_divisor`¶