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linalg · SolutionSpace

7 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.

linalg.SolutionSpace

A checked affine solution set over one field context.

class SolutionSpace

noether::linalg::SolutionSpace · Direct Rust declaration

pub struct SolutionSpace<R: Ring> { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/SolutionSpace@1

Executable usage: validation/unit/homalg/test_acceptance_workload.py:25 · validation/unit/homalg/test_acceptance_workload.py:9 · validation/unit/homalg/test_chain_complex.py:18 · validation/unit/homalg/test_chain_complex.py:40 · validation/unit/linalg/test_exact_linear_algebra_facade.py:30 · validation/unit/linalg/test_exact_linear_algebra_facade.py:79

linalg.SolutionSpace.ambient_dimension

Dimension of the ambient vector space.

(self) -> int

noether::linalg::SolutionSpace::ambient_dimension · Direct Rust declaration

pub fn ambient_dimension(&self) -> usize
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/SolutionSpace@1

Executable usage: validation/unit/commalg/test_kahler_differentials.py:98 · validation/unit/geometry/test_geometry_facade.py:23 · validation/unit/geometry/test_geometry_facade.py:46 · validation/unit/geometry/test_polytope_facade.py:30 · validation/unit/linalg/test_exact_linear_algebra_facade.py:152 · validation/unit/linalg/test_exact_linear_algebra_facade.py:153

linalg.SolutionSpace.basis

Deterministically ordered basis of the translation subspace.

(self) -> list[Vector[R]]

noether::linalg::SolutionSpace::basis · Direct Rust declaration

pub fn basis(&self) -> &[Vector<R>]
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/SolutionSpace@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

linalg.SolutionSpace.dimension

Dimension of the translation subspace.

(self) -> int

noether::linalg::SolutionSpace::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, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/SolutionSpace@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

linalg.SolutionSpace.particular

One point in the affine solution set.

(self) -> Vector[R]

noether::linalg::SolutionSpace::particular · Direct Rust declaration

pub fn particular(&self) -> &Vector<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/SolutionSpace@1

Executable usage: examples/03_exact_linear_systems.py:14 · examples/03_exact_linear_systems.py:16 · validation/unit/linalg/test_acceptance_workload.py:34 · validation/unit/linalg/test_acceptance_workload.py:48 · validation/unit/linalg/test_exact_linear_algebra_facade.py:147 · validation/unit/linalg/test_exact_linear_algebra_facade.py:202

linalg.SolutionSpace.ring

Coefficient field.

(self) -> R

noether::linalg::SolutionSpace::ring · Direct Rust declaration

pub fn ring(&self) -> &R
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/SolutionSpace@1

Executable usage: examples/11_qadic_witt_vectors.py:3 · examples/13_chinese_remainder_theorem.py:10 · examples/13_chinese_remainder_theorem.py:12 · examples/13_chinese_remainder_theorem.py:9 · examples/14_weighted_polynomials.py:10 · examples/14_weighted_polynomials.py:11

linalg.SolutionSpace.verify

Recheck the defining equations, completeness, and independence.

Detailed contract

Recheck the defining equations, dimension, and linear independence.

(self) -> None

noether::linalg::SolutionSpace::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, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/SolutionSpace@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