Noether

Noether is the deterministic computation layer for scientific mathematics: a typed Python package over a Rust engine and specialized exact-arithmetic kernels.

• Start with the guide

  Construct explicit coefficient contexts, compute, and verify results.

• Learn from 40 complete examples

  Every displayed program is executed by the ordinary test suite.

• Browse the API

  Search signatures, guarantees, evidence, failures, and Rust declarations.

• Understand contexts

  Rings, fields, maps, fixed precision, and mathematical identity.

A complete first calculation

This is not pseudocode. It is the checked source of Compute and verify a Bézout certificate over the integers.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Compute and verify a Bézout certificate over the integers."""

from noether import Z


def main() -> None:
    identity = Z.extended_gcd(987_654_321, 123_456_789)
    identity.verify(Z, 987_654_321, 123_456_789)
    assert (
        identity.coefficient_a * 987_654_321 + identity.coefficient_b * 123_456_789 == identity.gcd
    )
    print(identity)


if __name__ == "__main__":
    main()

Start from a mathematical task

• Solve exact systems over ℤ or ℤ/nℤ

  linalg.Matrix.solve_exact
  Run the complete example

• Compute a Smith normal form

  linalg.Matrix.smith_normal_form
  Run the complete example

• Construct and calculate in a finite field

  numthy.FiniteField.extension
  Run the complete example

• Compute a Gröbner basis and Hilbert invariants

  commalg.Ideal.groebner_basis
  Run the complete example

• Work with a native permutation group

  groups.PermutationGroup.from_generators
  Run the complete example

• Compute persistent homology

  topology.FilteredComplex.persistent_homology
  Run the complete example

• Build an exact convex hull

  geometry.convex_hull
  Run the complete example

• Compute with a finite higher category

  category.SteinerCategory.oriental
  Run the complete example

• Compute exact homology of a chain complex

  homalg.ChainComplex.homology_invariants
  Run the complete example

• Form Kähler differentials

  commalg.QuotientRing.kahler_differentials
  Run the complete example

• Compute in an exterior algebra

  homalg.FreeModule.exterior_algebra
  Run the complete example

Browse by domain

• category

  117 declarations
  Finite strong Steiner categories, cells, Gray products, mapping categories, and categorical homology presentations.

• combin

  191 declarations
  Finite graphs, trees, posets, matroids, tableaux, crystals, partitions, and symmetric functions.

• commalg

  1486 declarations
  Residue, polynomial, rational-function, quotient, and power-series rings together with ideals, schemes, local algebra, and intersection theory.

• geometry

  51 declarations
  Exact predicates, point clouds, triangulations, Euclidean balls, and exact rational polytopes.

• groups

  181 declarations
  Permutation groups, actions, conjugacy, characters, representations, and finite-group cohomology entry points.

• homalg

  638 declarations
  Free and presented modules, exact submodules, chain complexes, homology, resolutions, and algebraic constructions on modules.

• linalg

  370 declarations
  Dense, sparse, local, and MPI-distributed exact linear algebra, normal forms, solution modules, and lattices.

• numthy

  573 declarations
  Integer arithmetic, finite and local fields, number fields, orders, ideals, and elliptic curves.

• root

  148 declarations
  Shared coefficient contexts, exact scalar values, validity, errors, and installed-surface discovery.

• topology

  302 declarations
  Simplicial objects, filtrations, persistence, homology, cohomology operations, and symbolic Eilenberg–Mac Lane spaces.

For implementation details, open the architecture paper, the transport schemas, or the complete Rust rustdoc.