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commalg · RealSolutionBox

10 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.RealSolutionBox

One certified real solution box produced from a rational RUR.

class RealSolutionBox

noether::commalg::RealSolutionBox · Direct Rust declaration

pub struct RealSolutionBox { /* 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/RealSolutionBox@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:165 · validation/unit/commalg/test_polynomial_ideal_facade.py:173 · validation/unit/commalg/test_polynomial_ideal_facade.py:33

commalg.RealSolutionBox.coordinates

Return the coordinates recorded by this RealSolutionBox.

Detailed contract

Coordinate enclosures in source-variable order.

(self) -> list[RealBall]

noether::commalg::RealSolutionBox::coordinates · Direct Rust declaration

pub fn coordinates(&self) -> &[RealBall]
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/RealSolutionBox@1

Executable usage: validation/unit/commalg/test_base_change.py:114 · validation/unit/commalg/test_base_change.py:54 · validation/unit/commalg/test_base_change.py:62 · validation/unit/commalg/test_kahler_differentials.py:131 · validation/unit/commalg/test_polynomial_ideal_facade.py:171 · validation/unit/commalg/test_quotient_facade.py:142

commalg.RealSolutionBox.from_json

Deserialize and revalidate a RealSolutionBox from typed JSON.

Detailed contract

Deserialize and exactly reverify a typed certificate.

(cls, value: str) -> RealSolutionBox

noether::commalg::RealSolutionBox::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/RealSolutionBox@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.RealSolutionBox.multiplicity

Return the multiplicity recorded by this RealSolutionBox.

Detailed contract

Local multiplicity represented by this parameter root.

(self) -> int

noether::commalg::RealSolutionBox::multiplicity · Direct Rust declaration

pub fn multiplicity(&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/RealSolutionBox@1

Executable usage: validation/unit/combin/test_acceptance_workload.py:64 · validation/unit/commalg/test_local_algebra.py:65 · validation/unit/commalg/test_local_algebra.py:82 · validation/unit/commalg/test_polynomial_facade.py:126 · validation/unit/commalg/test_polynomial_facade.py:158 · validation/unit/commalg/test_polynomial_facade.py:96

commalg.RealSolutionBox.parameter

Return the parameter recorded by this RealSolutionBox.

Detailed contract

Exact isolating interval and multiplicity of the primitive parameter.

(self) -> RealRootIsolation

noether::commalg::RealSolutionBox::parameter · Direct Rust declaration

pub fn parameter(&self) -> &RealRootIsolation
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/RealSolutionBox@1

Executable usage: examples/26_elliptic_descent_functions.py:12 · examples/26_elliptic_descent_functions.py:15 · validation/unit/commalg/test_curve_facade.py:107 · validation/unit/commalg/test_curve_facade.py:110 · validation/unit/commalg/test_normalization_facade.py:35 · validation/unit/commalg/test_normalization_facade.py:36

commalg.RealSolutionBox.parameter_ball

Return the parameter ball recorded by this RealSolutionBox.

Detailed contract

Rigorous Arb enclosure of the primitive parameter.

(self) -> RealBall

noether::commalg::RealSolutionBox::parameter_ball · Direct Rust declaration

pub fn parameter_ball(&self) -> &RealBall
Contract Value
Guarantee rigorous enclosure
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/RealSolutionBox@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:170

commalg.RealSolutionBox.precision

Return the precision recorded by this RealSolutionBox.

Detailed contract

Arb working precision used for every returned ball.

(self) -> int

noether::commalg::RealSolutionBox::precision · Direct Rust declaration

pub fn precision(&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/RealSolutionBox@1

Executable usage: examples/10_padic_hensel_lifting.py:3 · examples/11_qadic_witt_vectors.py:3 · examples/31_local_fields_and_precision.py:3 · examples/31_local_fields_and_precision.py:34 · examples/31_local_fields_and_precision.py:38 · validation/unit/commalg/test_polynomial_facade.py:102

commalg.RealSolutionBox.representation

Return the representation recorded by this RealSolutionBox.

Detailed contract

Verified rational univariate representation used for this solution.

(self) -> RationalUnivariateRepresentation[RationalField]

noether::commalg::RealSolutionBox::representation · Direct Rust declaration

pub fn representation(&self) -> &RationalUnivariateRepresentation<RationalField>
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/RealSolutionBox@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:100 · validation/unit/commalg/test_polynomial_ideal_facade.py:101 · validation/unit/commalg/test_polynomial_ideal_facade.py:102 · validation/unit/commalg/test_polynomial_ideal_facade.py:103 · validation/unit/commalg/test_polynomial_ideal_facade.py:104 · validation/unit/commalg/test_polynomial_ideal_facade.py:105

commalg.RealSolutionBox.to_json

Serialize this RealSolutionBox to its canonical typed JSON representation.

Detailed contract

Serialize this certificate as typed JSON.

(self) -> str

noether::commalg::RealSolutionBox::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/RealSolutionBox@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.RealSolutionBox.verify

Replay and verify the mathematical evidence carried by this RealSolutionBox.

Detailed contract

Recompute the certified solution family and check exact identity.

(self) -> None

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

pub fn verify(&self) -> Result<(), RealSolutionError>
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/RealSolutionBox@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