Executable examples

These 40 complete programs use only Noether and the Python standard library. Every page displays the exact source executed by the ordinary test suite. Run the whole collection with make test-examples.

• Compute and verify a Bézout certificate over the integers

  examples/01_bezout_certificates.py
  Domains: root

• Certify a familiar identity with real and complex balls

  examples/02_ball_arithmetic.py
  Domains: root

• Solve an underdetermined rational system and verify its affine space

  examples/03_exact_linear_systems.py
  Domains: linalg, root

• Compute a fast Smith form without transformation matrices

  examples/04_smith_normal_form.py
  Domains: linalg, root

• Apply and multiply canonical sparse exact operators

  examples/05_sparse_exact_operators.py
  Domains: linalg, root

• Enumerate and verify the six shortest vectors of the A2 lattice

  examples/06_shortest_lattice_vectors.py
  Domains: linalg, root

• Factor an integer and construct a cyclotomic polynomial

  examples/07_integer_factorization.py
  Domains: numthy

• Calculate trace and norm in a concrete quadratic finite field

  examples/08_finite_field_extensions.py
  Domains: numthy

• Build and verify an isomorphism between two presentations of F_25

  examples/09_finite_field_isomorphisms.py
  Domains: numthy

• Lift a simple square root from F_5 to fixed-precision Z_5

  examples/10_padic_hensel_lifting.py
  Domains: commalg, numthy

• Use the fixed-precision Witt ring W(F_9)

  examples/11_qadic_witt_vectors.py
  Domains: numthy

• Split rational primes and factor an ideal in the Gaussian integers

  examples/12_number_field_ideals.py
  Domains: commalg, numthy, root

• Project and reconstruct a residue class by the Chinese remainder theorem

  examples/13_chinese_remainder_theorem.py
  Domains: commalg

• Decompose a polynomial by a nonstandard positive grading

  examples/14_weighted_polynomials.py
  Domains: commalg, root

• Recover the Hilbert invariants of the twisted cubic cone

  examples/15_groebner_twisted_cubic.py
  Domains: commalg, root

• Reduce and specialize an exact rational function

  examples/16_rational_functions.py
  Domains: commalg, root

• Calculate in a fixed-cap multivariate weighted power-series ring

  examples/17_weighted_power_series.py
  Domains: commalg, root

• Enumerate and certify rational points on a finite-field hyperbola

  examples/18_finite_field_schemes.py
  Domains: commalg, numthy

• Construct S4 natively and inspect its stabilizer chain consequences

  examples/19_permutation_groups.py
  Domains: groups

• Compute all conjugacy classes of A5 and verify orbit-stabilizer

  examples/20_conjugacy_classes.py
  Domains: groups

• Canonically label a cycle and compute its automorphism group

  examples/21_graph_symmetry.py
  Domains: combin

• Compute the Tutte polynomial of the uniform matroid U(2, 3)

  examples/22_matroids_and_tutte.py
  Domains: combin

• Decompose the tensor square of the type-A2 letter crystal

  examples/23_kashiwara_crystals.py
  Domains: combin

• See a one-cycle appear and die in a filtered triangle

  examples/24_homology_and_persistence.py
  Domains: numthy, topology

• Recover the exact face structure of a rational square

  examples/25_exact_geometry.py
  Domains: geometry

• Reproduce Fisher's order-five elliptic-function normalization

  examples/26_elliptic_descent_functions.py
  Domains: commalg, numthy, root

• Weight characters and second Chern classes for polynomial GL representations

  examples/27_weight_characters.py
  Domains: combin, groups

• Compose cells, compute polygraphic homology, and inspect the Street model

  examples/28_steiner_categories.py
  Domains: category

• Certify symmetries, realizations, and completions of integral lines

  examples/29_integral_line_configurations.py
  Domains: linalg

• Reduce a module element and certify a minimal free resolution

  examples/30_modules_and_resolutions.py
  Domains: commalg, homalg, linalg, root

• Use a ramified local field, its maximal order, and explicit precision loss

  examples/31_local_fields_and_precision.py
  Domains: commalg, numthy

• Reduce a finite Poset and its order complex with replayable evidence

  examples/32_poset_core_and_morse.py
  Domains: combin, root

• Compute the mod-two cohomology ring of RP2 and its Steenrod square

  examples/33_simplicial_cohomology_operations.py
  Domains: numthy, topology

• Certify a complete presentation of the mod-two cohomology of C2

  examples/34_finite_group_cohomology.py
  Domains: groups

• Compute exact additive and Cartan--Serre invariants of Eilenberg--Mac Lane spaces

  examples/35_eilenberg_maclane_spaces.py
  Domains: homalg, numthy, topology

• Normalize the cusp y^2=x^3 with exact conductor evidence

  examples/36_affine_normalization.py
  Domains: commalg, root

• Compute characteristic numbers of a smooth complete intersection

  examples/37_complete_intersection_theory.py
  Domains: commalg

• Exact affine systems and canonical submodules over Z and Z/nZ

  examples/38_exact_linear_systems.py
  Domains: commalg, homalg, linalg, root

• Differentiate exactly and distinguish a cusp's smooth and singular points

  examples/39_kahler_differentials.py
  Domains: commalg, root

• Compute in an exterior algebra and induce a linear map on two-forms

  examples/40_exterior_algebra.py
  Domains: commalg, homalg, linalg, root