commalg · MonomialIdeal¶

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

Worked example¶

The complete checked program Decompose a polynomial by a nonstandard positive grading uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Decompose a polynomial by a nonstandard positive grading."""

from noether import Z
from noether.commalg import PolynomialRing, WeightedGrading


def main() -> None:
    ring = PolynomialRing(Z, ["x", "y", "z"], grading=WeightedGrading([1, 2, 3]))
    x, y, z = ring.generators()
    sparse = ring.from_terms([([0, 0, 0], 1), ([2, 0, 0], 1), ([0, 1, 0], 1)])
    polynomial = sparse + z
    assert sparse * (x + y) == (1 + x**2 + y) * (x + y)
    components = polynomial.homogeneous_components()
    assert [degree for degree, _ in components] == [0, 2, 3]
    assert x**2 + y == components[1][1]
    print(f"weighted homogeneous degrees: {[degree for degree, _ in components]}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/14_weighted_polynomials.py.

`commalg.MonomialIdeal`¶

A canonical monomial ideal in one explicit polynomial-ring context.

PythonRust

class MonomialIdeal

noether::commalg::MonomialIdeal · Direct Rust declaration

pub struct MonomialIdeal<R: CommutativeRing> { /* 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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:14 · validation/unit/commalg/test_monomial_ideal_facade.py:25 · validation/unit/commalg/test_monomial_ideal_facade.py:69 · validation/unit/commalg/test_monomial_ideal_facade.py:70 · validation/unit/commalg/test_monomial_ideal_facade.py:99

`commalg.MonomialIdeal.add`¶

Add another compatible value to this value.

PythonRust

(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal · Python projection

Python projection convenience over this Rust mathematical type

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/MonomialIdeal@1

Executable usage: validation/unit/infra/test_discovery.py:211 · validation/unit/infra/test_public_api_index.py:293

`commalg.MonomialIdeal.contains`¶

Test whether the supplied value belongs to this object.

PythonRust

(self, exponent: Sequence[int]) -> bool

noether::commalg::MonomialIdeal · Python projection

Python projection convenience over this Rust mathematical type

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/MonomialIdeal@1

Executable usage: validation/unit/infra/test_discovery.py:200

`commalg.MonomialIdeal.mul`¶

Multiply this value by another compatible value.

PythonRust

(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal · Python projection

Python projection convenience over this Rust mathematical type

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/MonomialIdeal@1

Executable usage: validation/unit/infra/test_public_api_index.py:214

`commalg.MonomialIdeal.pow`¶

Raise this value to the supplied power.

PythonRust

(self, exponent: int, modulus: None=None) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal · Python projection

Python projection convenience over this Rust mathematical type

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/MonomialIdeal@1

`commalg.MonomialIdeal.colon`¶

Compute colon for this MonomialIdeal.

Detailed contract

Exact ideal quotient self : other.

PythonRust

(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::colon · Direct Rust declaration

pub fn colon(&self, other: &Self) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:34 · validation/unit/commalg/test_polynomial_ideal_facade.py:346 · validation/unit/commalg/test_polynomial_ideal_facade.py:347 · validation/unit/commalg/test_polynomial_ideal_facade.py:349 · validation/corpus/commalg/monomial-ideal-colon-witnesses.py:22 · validation/corpus/commalg/monomial-ideal-colon-witnesses.py:23

`commalg.MonomialIdeal.colon_monomial`¶

Compute colon monomial for this MonomialIdeal.

Detailed contract

Colon by one monomial.

PythonRust

(self, exponent: Sequence[int]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::colon_monomial · Direct Rust declaration

pub fn colon_monomial( &self, exponent: &[Integer], ) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

`commalg.MonomialIdeal.contains_monomial`¶

Compute contains monomial for this MonomialIdeal.

Detailed contract

Decide membership of one monomial exponent vector.

PythonRust

(self, exponent: Sequence[int]) -> bool

noether::commalg::MonomialIdeal::contains_monomial · Direct Rust declaration

pub fn contains_monomial( &self, exponent: &[Integer], ) -> Result<bool, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: validation/corpus/commalg/cycle-vertex-cover-power.py:16

`commalg.MonomialIdeal.degree`¶

Compute degree for this MonomialIdeal.

Detailed contract

Standard-graded multiplicity of R/I.

PythonRust

(self) -> int

noether::commalg::MonomialIdeal::degree · Direct Rust declaration

pub fn degree(&self) -> Result<Integer, GroebnerError>

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/MonomialIdeal@1

Executable usage: examples/14_weighted_polynomials.py:16 · examples/14_weighted_polynomials.py:18 · examples/15_groebner_twisted_cubic.py:15 · examples/15_groebner_twisted_cubic.py:28 · examples/31_local_fields_and_precision.py:37 · examples/33_simplicial_cohomology_operations.py:26

`commalg.MonomialIdeal.dimension`¶

Compute dimension for this MonomialIdeal.

Detailed contract

Krull dimension of R/I, with -1 for the unit ideal.

PythonRust

(self) -> int

noether::commalg::MonomialIdeal::dimension · Direct Rust declaration

pub fn dimension(&self) -> Result<i64, GroebnerError>

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/MonomialIdeal@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

`commalg.MonomialIdeal.finite_standard_monomials`¶

Compute finite standard monomials for this MonomialIdeal.

Detailed contract

Complete standard-monomial basis of a zero-dimensional quotient.

PythonRust

(self) -> list[list[int]]

noether::commalg::MonomialIdeal::finite_standard_monomials · Direct Rust declaration

pub fn finite_standard_monomials( &self, ) -> Result<Vec<Vec<Integer>>, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:46

`commalg.MonomialIdeal.from_json`¶

Deserialize and revalidate a MonomialIdeal from typed JSON.

Detailed contract

Deserialize and revalidate a canonical monomial ideal.

PythonRust

(cls, value: str) -> MonomialIdeal[CoefficientRing]

noether::commalg::MonomialIdeal::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>where R: for<'de> Deserialize<'de>,

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/MonomialIdeal@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.MonomialIdeal.hilbert_numerator`¶

Compute hilbert numerator for this MonomialIdeal.

Detailed contract

Numerator K(t) in the standard-graded identity H(R/I,t)=K(t)/(1-t)^n.

PythonRust

(self) -> list[int]

noether::commalg::MonomialIdeal::hilbert_numerator · Direct Rust declaration

pub fn hilbert_numerator(&self) -> Result<Vec<Integer>, GroebnerError>

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/MonomialIdeal@1

Executable usage: examples/15_groebner_twisted_cubic.py:16 · validation/unit/commalg/test_monomial_ideal_facade.py:47 · validation/unit/commalg/test_monomial_ideal_facade.py:50 · validation/unit/commalg/test_polynomial_ideal_facade.py:203 · validation/unit/commalg/test_polynomial_ideal_facade.py:205 · validation/unit/commalg/test_quotient_facade.py:126

`commalg.MonomialIdeal.hilbert_polynomial`¶

Compute hilbert polynomial for this MonomialIdeal.

Detailed contract

Standard Hilbert polynomial in the canonical variable m.

PythonRust

(self) -> Polynomial[RationalField]

noether::commalg::MonomialIdeal::hilbert_polynomial · Direct Rust declaration

pub fn hilbert_polynomial( &self, ) -> Result<Polynomial<RationalField>, GroebnerError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:54 · validation/unit/commalg/test_polynomial_ideal_facade.py:212 · validation/unit/commalg/test_polynomial_ideal_facade.py:223 · validation/unit/commalg/test_polynomial_ideal_facade.py:246 · validation/unit/commalg/test_quotient_facade.py:129 · validation/unit/commalg/test_scheme_facade.py:38

`commalg.MonomialIdeal.hilbert_series`¶

Compute hilbert series for this MonomialIdeal.

Detailed contract

Exact Hilbert series in the group algebra of the ring grading.

PythonRust

(self) -> HilbertSeries

noether::commalg::MonomialIdeal::hilbert_series · Direct Rust declaration

pub fn hilbert_series(&self) -> Result<HilbertSeries, GroebnerError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:48 · validation/unit/commalg/test_monomial_ideal_facade.py:86 · validation/unit/commalg/test_polynomial_ideal_facade.py:204 · validation/unit/commalg/test_quotient_facade.py:125 · validation/unit/homalg/test_module_facade.py:261 · validation/unit/homalg/test_module_facade.py:502

`commalg.MonomialIdeal.intersection`¶

Compute intersection for this MonomialIdeal.

Detailed contract

Intersection of two monomial ideals through pairwise least common multiples.

PythonRust

(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::intersection · Direct Rust declaration

pub fn intersection(&self, other: &Self) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: examples/37_complete_intersection_theory.py:3 · examples/39_kahler_differentials.py:37 · validation/unit/commalg/test_intersection_theory.py:100 · validation/unit/commalg/test_intersection_theory.py:102 · validation/unit/commalg/test_intersection_theory.py:3 · validation/unit/commalg/test_monomial_ideal_facade.py:33

`commalg.MonomialIdeal.is_unit`¶

Test whether this MonomialIdeal is unit.

Detailed contract

Whether this is the unit ideal.

PythonRust

(self) -> bool

noether::commalg::MonomialIdeal::is_unit · Direct Rust declaration

pub fn is_unit(&self) -> bool

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_local_algebra.py:44 · validation/unit/commalg/test_local_algebra.py:59 · validation/unit/commalg/test_prime_localization.py:25 · validation/unit/commalg/test_prime_localization.py:26 · validation/unit/commalg/test_prime_localization.py:47 · validation/unit/commalg/test_quotient_facade.py:51

`commalg.MonomialIdeal.is_zero`¶

Test whether this MonomialIdeal is zero.

Detailed contract

Whether this is the zero ideal.

PythonRust

(self) -> bool

noether::commalg::MonomialIdeal::is_zero · Direct Rust declaration

pub fn is_zero(&self) -> bool

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/MonomialIdeal@1

Executable usage: examples/17_weighted_power_series.py:16 · examples/17_weighted_power_series.py:17 · examples/39_kahler_differentials.py:20 · validation/unit/commalg/test_graded_commutative_facade.py:26 · validation/unit/commalg/test_kahler_differentials.py:113 · validation/unit/commalg/test_kahler_differentials.py:38

`commalg.MonomialIdeal.minimal_generators`¶

Canonical minimal exponent antichain.

Detailed contract

Minimal monomial generators as exact exponent vectors.

PythonRust

(self) -> list[list[int]]

noether::commalg::MonomialIdeal::minimal_generators · Direct Rust declaration

pub fn minimal_generators(&self) -> &[Vec<Integer>]

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:27 · validation/unit/commalg/test_monomial_ideal_facade.py:31 · validation/unit/commalg/test_monomial_ideal_facade.py:32 · validation/unit/commalg/test_monomial_ideal_facade.py:33 · validation/unit/commalg/test_monomial_ideal_facade.py:34 · validation/unit/commalg/test_monomial_ideal_facade.py:35

`commalg.MonomialIdeal.multigraded_hilbert_numerator`¶

Compute multigraded hilbert numerator for this MonomialIdeal.

Detailed contract

Hilbert numerator in the group algebra of the ring’s full grading. This K-polynomial is defined without a positivity hypothesis. When the grading has finite pieces, adjoining denominator factors 1 - T^d for the variable degrees d gives the exact multigraded Hilbert series of the quotient.

PythonRust

(self) -> list[tuple[Degree, int]]

noether::commalg::MonomialIdeal::multigraded_hilbert_numerator · Direct Rust declaration

pub fn multigraded_hilbert_numerator( &self, ) -> Result<Vec<(Degree, Integer)>, GroebnerError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:80 · validation/unit/commalg/test_monomial_ideal_facade.py:86 · validation/unit/homalg/test_module_facade.py:257 · validation/unit/homalg/test_module_facade.py:263

`commalg.MonomialIdeal.pow`¶

Compute pow for this MonomialIdeal.

Detailed contract

Nonnegative ideal power, with I^0 = (1).

PythonRust

(self, exponent: int) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::pow · Direct Rust declaration

pub fn pow(&self, exponent: u64) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: examples/17_weighted_power_series.py:18 · validation/unit/commalg/test_polynomial_ideal_facade.py:368 · validation/unit/commalg/test_polynomial_ideal_facade.py:369 · validation/unit/commalg/test_polynomial_ideal_facade.py:370 · validation/unit/commalg/test_polynomial_ideal_facade.py:373 · validation/unit/commalg/test_polynomial_ideal_facade.py:381

`commalg.MonomialIdeal.product`¶

Compute product for this MonomialIdeal.

Detailed contract

Product of two monomial ideals.

PythonRust

(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::product · Direct Rust declaration

pub fn product(&self, other: &Self) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: examples/33_simplicial_cohomology_operations.py:27 · validation/unit/combin/test_acceptance_workload.py:41 · validation/unit/combin/test_acceptance_workload.py:43 · validation/unit/combin/test_acceptance_workload.py:44 · validation/unit/combin/test_acceptance_workload.py:45 · validation/unit/commalg/test_polynomial_facade.py:234

`commalg.MonomialIdeal.radical`¶

Compute radical for this MonomialIdeal.

Detailed contract

Radical, obtained by replacing every positive exponent by one.

PythonRust

(self) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::radical · Direct Rust declaration

pub fn radical(&self) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:35 · validation/unit/commalg/test_polynomial_ideal_facade.py:1000 · validation/unit/commalg/test_polynomial_ideal_facade.py:1023 · validation/unit/commalg/test_polynomial_ideal_facade.py:1034 · validation/unit/commalg/test_polynomial_ideal_facade.py:1035 · validation/unit/commalg/test_polynomial_ideal_facade.py:1036

`commalg.MonomialIdeal.ring`¶

Ambient polynomial-ring context.

PythonRust

(self) -> PolynomialRing[C]

noether::commalg::MonomialIdeal::ring · Direct Rust declaration

pub fn ring(&self) -> &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/MonomialIdeal@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

`commalg.MonomialIdeal.saturate`¶

Compute saturate for this MonomialIdeal.

Detailed contract

Saturation by all powers of another monomial ideal.

PythonRust

(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::saturate · Direct Rust declaration

pub fn saturate(&self, other: &Self) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:36 · validation/unit/commalg/test_polynomial_ideal_facade.py:348 · validation/unit/commalg/test_polynomial_ideal_facade.py:350 · validation/corpus/commalg/dingdong-weighted-critical-ideal.py:23 · validation/corpus/commalg/foliation-singular-scheme-saturation.py:33 · validation/corpus/commalg/singular-locus-saturation.py:15

`commalg.MonomialIdeal.saturate_monomial`¶

Compute saturate monomial for this MonomialIdeal.

Detailed contract

Saturation by all powers of one monomial.

PythonRust

(self, exponent: Sequence[int]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::saturate_monomial · Direct Rust declaration

pub fn saturate_monomial( &self, exponent: &[Integer], ) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

`commalg.MonomialIdeal.standard_monomials_of_degree`¶

Compute standard monomials of degree for this MonomialIdeal.

Detailed contract

Standard monomials of one exact total degree, with an explicit output cap.

PythonRust

(self, degree: int, *, max_count: int) -> list[list[int]]

noether::commalg::MonomialIdeal::standard_monomials_of_degree · Direct Rust declaration

pub fn standard_monomials_of_degree( &self, degree: usize, maximum_count: usize, ) -> Result<Vec<Vec<Integer>>, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:45 · validation/unit/commalg/test_monomial_ideal_facade.py:58 · validation/corpus/commalg/octic-quotient-degree-eight.py:14

`commalg.MonomialIdeal.sum`¶

Compute sum for this MonomialIdeal.

Detailed contract

Sum of two monomial ideals.

PythonRust

(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]

noether::commalg::MonomialIdeal::sum · Direct Rust declaration

pub fn sum(&self, other: &Self) -> Result<Self, MonomialIdealError>

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/MonomialIdeal@1

Executable usage: examples/32_poset_core_and_morse.py:18 · validation/unit/combin/test_acceptance_workload.py:64 · validation/unit/combinatorics/test_boundary.py:31 · validation/unit/commalg/test_acceptance_workload.py:137 · validation/unit/commalg/test_monomial_ideal_facade.py:94 · validation/unit/commalg/test_polynomial_ideal_facade.py:334

`commalg.MonomialIdeal.support`¶

Compute support for this MonomialIdeal.

Detailed contract

Sorted variables occurring in a minimal generator.

PythonRust

(self) -> list[int]

noether::commalg::MonomialIdeal::support · Direct Rust declaration

pub fn support(&self) -> Vec<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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:30 · validation/unit/commalg/test_polynomial_ideal_facade.py:316 · validation/unit/commalg/test_polynomial_ideal_facade.py:317 · validation/unit/commalg/test_polynomial_ideal_facade.py:318 · validation/unit/commalg/test_polynomial_ideal_facade.py:319 · validation/unit/commalg/test_polynomial_ideal_facade.py:320

`commalg.MonomialIdeal.to_ideal`¶

Convert to a polynomial ideal in a currently projected Groebner field.

Detailed contract

Convert to the canonical polynomial ideal in the same ring.

PythonRust

(self: MonomialIdeal[_ProjectedIdealFieldContext]) -> Ideal[PolynomialRing[_ProjectedIdealFieldContext]]

noether::commalg::MonomialIdeal::to_ideal · Direct Rust declaration

pub fn to_ideal(&self) -> Result<Ideal<PolynomialRing<R>>, GroebnerError>

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/MonomialIdeal@1

Executable usage: validation/unit/commalg/test_monomial_ideal_facade.py:55 · validation/unit/commalg/test_monomial_ideal_facade.py:71 · validation/unit/commalg/test_monomial_ideal_facade.py:72 · validation/unit/commalg/test_polynomial_ideal_facade.py:553

`commalg.MonomialIdeal.to_json`¶

Serialize this MonomialIdeal to its canonical typed JSON representation.

Detailed contract

Serialize the complete ring context and canonical exponent antichain.

PythonRust

(self) -> str

noether::commalg::MonomialIdeal::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>where R: Serialize,

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/MonomialIdeal@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.MonomialIdeal.verify`¶

Replay and verify the mathematical evidence carried by this MonomialIdeal.

Detailed contract

Recheck the canonical minimal-antichain representation.

PythonRust

(self) -> None

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

pub fn verify(&self) -> Result<(), MonomialIdealError>

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/MonomialIdeal@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

commalg · MonomialIdeal¶

Worked example¶

commalg.MonomialIdeal¶

commalg.MonomialIdeal.__add__¶

commalg.MonomialIdeal.__contains__¶

commalg.MonomialIdeal.__mul__¶

commalg.MonomialIdeal.__pow__¶

commalg.MonomialIdeal.colon¶

commalg.MonomialIdeal.colon_monomial¶

commalg.MonomialIdeal.contains_monomial¶

commalg.MonomialIdeal.degree¶

commalg.MonomialIdeal.dimension¶

commalg.MonomialIdeal.finite_standard_monomials¶

commalg.MonomialIdeal.from_json¶

commalg.MonomialIdeal.hilbert_numerator¶

commalg.MonomialIdeal.hilbert_polynomial¶

commalg.MonomialIdeal.hilbert_series¶

commalg.MonomialIdeal.intersection¶

commalg.MonomialIdeal.is_unit¶

commalg.MonomialIdeal.is_zero¶

commalg.MonomialIdeal.minimal_generators¶

commalg.MonomialIdeal.multigraded_hilbert_numerator¶

commalg.MonomialIdeal.pow¶

commalg.MonomialIdeal.product¶

commalg.MonomialIdeal.radical¶

commalg.MonomialIdeal.ring¶

commalg.MonomialIdeal.saturate¶

commalg.MonomialIdeal.saturate_monomial¶

commalg.MonomialIdeal.standard_monomials_of_degree¶

commalg.MonomialIdeal.sum¶

commalg.MonomialIdeal.support¶

commalg.MonomialIdeal.to_ideal¶

commalg.MonomialIdeal.to_json¶

commalg.MonomialIdeal.verify¶

`commalg.MonomialIdeal`¶

`commalg.MonomialIdeal.add`¶

`commalg.MonomialIdeal.contains`¶

`commalg.MonomialIdeal.mul`¶

`commalg.MonomialIdeal.pow`¶

`commalg.MonomialIdeal.colon`¶

`commalg.MonomialIdeal.colon_monomial`¶

`commalg.MonomialIdeal.contains_monomial`¶

`commalg.MonomialIdeal.degree`¶

`commalg.MonomialIdeal.dimension`¶

`commalg.MonomialIdeal.finite_standard_monomials`¶

`commalg.MonomialIdeal.from_json`¶

`commalg.MonomialIdeal.hilbert_numerator`¶

`commalg.MonomialIdeal.hilbert_polynomial`¶

`commalg.MonomialIdeal.hilbert_series`¶

`commalg.MonomialIdeal.intersection`¶

`commalg.MonomialIdeal.is_unit`¶

`commalg.MonomialIdeal.is_zero`¶

`commalg.MonomialIdeal.minimal_generators`¶

`commalg.MonomialIdeal.multigraded_hilbert_numerator`¶

`commalg.MonomialIdeal.pow`¶

`commalg.MonomialIdeal.product`¶

`commalg.MonomialIdeal.radical`¶

`commalg.MonomialIdeal.ring`¶

`commalg.MonomialIdeal.saturate`¶

`commalg.MonomialIdeal.saturate_monomial`¶

`commalg.MonomialIdeal.standard_monomials_of_degree`¶

`commalg.MonomialIdeal.sum`¶

`commalg.MonomialIdeal.support`¶

`commalg.MonomialIdeal.to_ideal`¶

`commalg.MonomialIdeal.to_json`¶

`commalg.MonomialIdeal.verify`¶