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commalg · EquidimensionalHull

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

Worked example

The complete checked program Differentiate exactly and distinguish a cusp's smooth and singular points uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Differentiate exactly and distinguish a cusp's smooth and singular points."""

from noether import Q
from noether.commalg import KahlerDifferentials, PolynomialRing


def main() -> None:
    ring = PolynomialRing(Q, ["x", "y"])
    x, y = ring.generators()
    relation = y**2 - x**3
    cusp = ring.quotient(ring.ideal([relation]))
    omega = cusp.kahler_differentials()

    assert isinstance(omega, KahlerDifferentials)
    dx = omega.differentiate(cusp(x))
    dy = omega.differentiate(cusp(y))
    assert omega.differentiate(cusp(x * y)) == dx * cusp(y) + dy * cusp(x)
    assert omega.differentiate(cusp(relation)).is_zero()

    singular = cusp.cotangent_space_at([0, 0])
    smooth = cusp.cotangent_space_at([1, 1])
    assert singular.embedding_dimension == 2
    assert smooth.embedding_dimension == 1
    assert singular.presentation_matrix == singular.evaluated_jacobian.transpose()

    complex = cusp.cotangent_complex()
    assert complex.codimension == 1
    assert (complex.amplitude.lowest_degree, complex.amplitude.highest_degree) == (-1, 0)
    assert complex.hypothesis.equation_count == 1
    complex.verify()

    print(
        "cusp cotangent dimensions: "
        f"singular={singular.embedding_dimension}, smooth={smooth.embedding_dimension}; "
        "complete-intersection amplitude=[-1,0]"
    )


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/39_kahler_differentials.py.

commalg.EquidimensionalHull

Verified scheme-theoretic top-dimensional hull of a polynomial ideal.

class EquidimensionalHull

noether::commalg::EquidimensionalHull · Direct Rust declaration

pub struct EquidimensionalHull<R: Field> { /* 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/EquidimensionalHull@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:16 · validation/unit/commalg/test_polynomial_ideal_facade.py:650 · validation/unit/commalg/test_polynomial_ideal_facade.py:661 · validation/unit/commalg/test_polynomial_ideal_facade.py:666

commalg.EquidimensionalHull.associated_primes

Return the associated primes recorded by this EquidimensionalHull.

Detailed contract

Complete EHV evidence from which the hull is extracted.

(self) -> AssociatedPrimeSet[C]

noether::commalg::EquidimensionalHull::associated_primes · Direct Rust declaration

pub fn associated_primes(&self) -> &AssociatedPrimeSet<R>
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/EquidimensionalHull@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1271 · validation/unit/commalg/test_polynomial_ideal_facade.py:566 · validation/unit/commalg/test_polynomial_ideal_facade.py:577 · validation/unit/commalg/test_polynomial_ideal_facade.py:623 · validation/unit/commalg/test_polynomial_ideal_facade.py:654 · validation/unit/commalg/test_polynomial_ideal_facade.py:694

commalg.EquidimensionalHull.codimension

Return the codimension recorded by this EquidimensionalHull.

Detailed contract

Codimension of the source and its top-dimensional hull.

(self) -> int

noether::commalg::EquidimensionalHull::codimension · Direct Rust declaration

pub const fn codimension(&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/EquidimensionalHull@1

Executable usage: examples/39_kahler_differentials.py:29 · validation/unit/commalg/test_intersection_theory.py:31 · validation/unit/commalg/test_intersection_theory.py:32 · validation/unit/commalg/test_intersection_theory.py:50 · validation/unit/commalg/test_kahler_differentials.py:100 · validation/unit/commalg/test_kahler_differentials.py:121

commalg.EquidimensionalHull.from_json

Deserialize and revalidate a EquidimensionalHull from typed JSON.

Detailed contract

Deserialize and exactly reverify an equidimensional hull.

(cls, value: str) -> EquidimensionalHull[RationalField | FiniteField]

noether::commalg::EquidimensionalHull::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/EquidimensionalHull@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.EquidimensionalHull.ideal

Return the ideal recorded by this EquidimensionalHull.

Detailed contract

Intersection of the source’s primary components of maximal dimension.

(self) -> Ideal[PolynomialRing[C]]

noether::commalg::EquidimensionalHull::ideal · Direct Rust declaration

pub fn ideal(&self) -> &Ideal<PolynomialRing<R>>
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/EquidimensionalHull@1

Executable usage: examples/12_number_field_ideals.py:17 · examples/12_number_field_ideals.py:18 · examples/12_number_field_ideals.py:19 · examples/12_number_field_ideals.py:3 · examples/15_groebner_twisted_cubic.py:12 · examples/15_groebner_twisted_cubic.py:13

commalg.EquidimensionalHull.minimal_primes

Return the minimal primes recorded by this EquidimensionalHull.

Detailed contract

Certified minimal primes of the hull, absent only for the unit ideal.

(self) -> MinimalPrimeDecomposition[C] | None

noether::commalg::EquidimensionalHull::minimal_primes · Direct Rust declaration

pub fn minimal_primes(&self) -> Option<&MinimalPrimeDecomposition<R>>
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/EquidimensionalHull@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:69 · validation/unit/commalg/test_acceptance_workload.py:70 · validation/unit/commalg/test_acceptance_workload.py:75 · validation/unit/commalg/test_affine_normalization_facade.py:183 · validation/unit/commalg/test_polynomial_ideal_facade.py:1002 · validation/unit/commalg/test_polynomial_ideal_facade.py:1039

commalg.EquidimensionalHull.prime_certificates

Return the prime certificates recorded by this EquidimensionalHull.

Detailed contract

Canonically ordered top-dimensional prime certificates.

(self) -> list[PolynomialPrimeCertificate[C]]

noether::commalg::EquidimensionalHull::prime_certificates · Direct Rust declaration

pub fn prime_certificates(&self) -> &[PolynomialPrimeCertificate<R>]
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/EquidimensionalHull@1

Executable usage: validation/unit/commalg/test_polynomial_ideal_facade.py:1006 · validation/unit/commalg/test_polynomial_ideal_facade.py:1041 · validation/unit/commalg/test_polynomial_ideal_facade.py:1057 · validation/unit/commalg/test_polynomial_ideal_facade.py:1075 · validation/unit/commalg/test_polynomial_ideal_facade.py:1140 · validation/unit/commalg/test_polynomial_ideal_facade.py:1143

commalg.EquidimensionalHull.source

Return the source recorded by this EquidimensionalHull.

Detailed contract

Original polynomial ideal before lower-dimensional components are removed.

(self) -> Ideal[PolynomialRing[C]]

noether::commalg::EquidimensionalHull::source · Direct Rust declaration

pub fn source(&self) -> &Ideal<PolynomialRing<R>>
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/EquidimensionalHull@1

Executable usage: examples/09_finite_field_isomorphisms.py:11 · examples/09_finite_field_isomorphisms.py:13 · examples/09_finite_field_isomorphisms.py:9 · examples/30_modules_and_resolutions.py:34 · examples/30_modules_and_resolutions.py:37 · validation/unit/category/test_acceptance_workload.py:25

commalg.EquidimensionalHull.theorem

Return the theorem recorded by this EquidimensionalHull.

Detailed contract

Exact theorem identifying the retained ideal.

(self) -> str

noether::commalg::EquidimensionalHull::theorem · Direct Rust declaration

pub const fn theorem(&self) -> &'static str
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/EquidimensionalHull@1

Executable usage: examples/13_chinese_remainder_theorem.py:3 · validation/unit/commalg/test_polynomial_ideal_facade.py:1150 · validation/unit/commalg/test_polynomial_ideal_facade.py:507 · validation/unit/commalg/test_polynomial_ideal_facade.py:583 · validation/unit/commalg/test_polynomial_ideal_facade.py:659 · validation/unit/homalg/test_module_facade.py:509

commalg.EquidimensionalHull.to_json

Serialize this EquidimensionalHull to its canonical typed JSON representation.

Detailed contract

Serialize the source, hull, and complete Ext-annihilator evidence.

(self) -> str

noether::commalg::EquidimensionalHull::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/EquidimensionalHull@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.EquidimensionalHull.verify

Replay and verify the mathematical evidence carried by this EquidimensionalHull.

Detailed contract

Independently replay the complete Ext-annihilator evidence.

(self) -> None

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

pub fn verify(&self) -> Result<(), AssociatedPrimeError>
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/EquidimensionalHull@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