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root · Types and values

3 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example

The complete checked program Solve an underdetermined rational system and verify its affine space uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Solve an underdetermined rational system and verify its affine space."""

from noether import Q
from noether.linalg import Matrix, Vector


def main() -> None:
    matrix = Matrix.from_rows(Q, [[1, 2, 3], [2, 4, 6]])
    right_hand_side = Vector.from_values(Q, [1, 2])
    solutions = matrix.solve(right_hand_side)
    solutions.verify()
    assert matrix @ solutions.particular == right_hand_side
    assert solutions.dimension == 2
    print(f"particular={solutions.particular}; nullity={solutions.dimension}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/03_exact_linear_systems.py.

Q

The canonical exact rational field.

Q: RationalField

noether::Q · Direct Rust declaration

pub const Q: RationalField;
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded

Executable usage: examples/03_exact_linear_systems.py:10 · examples/03_exact_linear_systems.py:11 · examples/03_exact_linear_systems.py:5 · examples/12_number_field_ideals.py:11 · examples/12_number_field_ideals.py:5 · examples/15_groebner_twisted_cubic.py:10

Z

The canonical arbitrary-precision integer ring.

Z: IntegerRing

noether::Z · Direct Rust declaration

pub const Z: IntegerRing;
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:5 · examples/01_bezout_certificates.py:9 · examples/04_smith_normal_form.py:10 · examples/04_smith_normal_form.py:5 · examples/05_sparse_exact_operators.py:11

__version__

The installed Noether package version.

__version__: str

noether::version · Python projection

Python projection convenience; no separate Rust item
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded

Executable usage: validation/unit/infra/test_discovery.py:253 · validation/unit/infra/test_discovery.py:300 · validation/unit/infra/test_discovery.py:301 · validation/bench/magma_release.py:2153 · validation/bench/sage_exterior_algebra.py:104