commalg · MonomialMap

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

Worked example

The complete checked program Apply and multiply canonical sparse exact operators uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Apply and multiply canonical sparse exact operators."""

from noether import Z
from noether.linalg import SparseMatrix, Vector


def main() -> None:
    operator = SparseMatrix.from_triplets(
        Z,
        (3, 3),
        [(0, 0, 2), (0, 2, -1), (1, 1, 3), (2, 0, 4)],
    )
    vector = Vector.from_values(Z, [5, 7, 11])
    image = operator.apply(vector)
    assert image.to_values() == [-1, 21, 20]
    assert (operator @ operator).nnz == 4
    print(f"A v = {image.to_values()}; nnz(A^2) = {(operator @ operator).nnz}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/05_sparse_exact_operators.py.

commalg.MonomialMap

A checked monomial endomorphism on an ordered variable list.

PythonRust

MonomialMap(images: Iterable[Iterable[int]])

noether::commalg::MonomialMap · Direct Rust declaration

pub struct MonomialMap { /* 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/MonomialMap@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:12 · validation/unit/commalg/test_polynomial_facade.py:195 · validation/unit/commalg/test_polynomial_facade.py:200 · validation/unit/commalg/test_polynomial_facade.py:201 · validation/unit/commalg/test_polynomial_facade.py:204 · validation/unit/commalg/test_polynomial_facade.py:206

commalg.MonomialMap.apply

Apply this MonomialMap to the supplied value.

Detailed contract

Transform one arbitrary-precision nonnegative exponent vector.

PythonRust

(self, exponent: Iterable[int]) -> list[int]

noether::commalg::MonomialMap::apply · Direct Rust declaration

pub fn apply( &self, exponent: &[Integer], ) -> Result<Vec<Integer>, MonomialMapError>

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/MonomialMap@1

Executable usage: examples/05_sparse_exact_operators.py:16 · validation/unit/commalg/test_local_algebra.py:94 · validation/unit/commalg/test_local_algebra.py:97 · validation/unit/commalg/test_polynomial_facade.py:199 · validation/unit/commalg/test_quotient_facade.py:186 · validation/unit/groups/test_weight_characters.py:39

commalg.MonomialMap.arity

Return the arity recorded by this MonomialMap.

Detailed contract

Number of ordered source and target variables.

PythonRust

(self) -> int

noether::commalg::MonomialMap::arity · Direct Rust declaration

pub fn arity(&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/MonomialMap@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:197 · validation/unit/commalg/test_polynomial_facade.py:205 · validation/unit/commalg/test_polynomial_ideal_facade.py:259 · validation/unit/commalg/test_power_series_facade.py:65 · validation/bench/groebner_compare.py:120 · validation/bench/groebner_compare.py:121

commalg.MonomialMap.compose

Compose this MonomialMap with another compatible map.

Detailed contract

Compose self after inner using exact exponent arithmetic.

PythonRust

(self, inner: MonomialMap) -> MonomialMap

noether::commalg::MonomialMap::compose · Direct Rust declaration

pub fn compose(&self, inner: &Self) -> Result<Self, MonomialMapError>

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/MonomialMap@1

Executable usage: validation/unit/commalg/test_context_tower_workload.py:3 · validation/unit/commalg/test_polynomial_facade.py:200 · validation/unit/commalg/test_power_series_facade.py:151 · validation/unit/commalg/test_power_series_facade.py:245 · validation/unit/commalg/test_power_series_facade.py:248 · validation/unit/groups/test_permutation_group_facade.py:41

commalg.MonomialMap.diagonal_power

Construct a MonomialMap using diagonal power.

Detailed contract

Diagonal substitution x_i -> x_i^power on every variable.

PythonRust

(cls, arity: int, power: int) -> MonomialMap

noether::commalg::MonomialMap::diagonal_power · Direct Rust declaration

pub fn diagonal_power( arity: usize, power: Integer, ) -> Result<Self, MonomialMapError>

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/MonomialMap@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:247 · validation/unit/commalg/test_power_series_facade.py:280 · validation/unit/commalg/test_power_series_facade.py:287

commalg.MonomialMap.from_json

Deserialize and revalidate a MonomialMap from typed JSON.

Detailed contract

Deserialize and revalidate every exponent vector.

PythonRust

(cls, value: str) -> MonomialMap

noether::commalg::MonomialMap::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/MonomialMap@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.MonomialMap.identity

Construct a MonomialMap using identity.

Detailed contract

Identity monomial map on arity ordered variables.

PythonRust

(cls, arity: int) -> MonomialMap

noether::commalg::MonomialMap::identity · Direct Rust declaration

pub fn identity(arity: usize) -> Self

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/MonomialMap@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:12 · examples/01_bezout_certificates.py:14 · examples/01_bezout_certificates.py:9 · examples/02_ball_arithmetic.py:3 · examples/distributed/f2_linear_algebra.py:20

commalg.MonomialMap.images

Return the images recorded by this MonomialMap.

Detailed contract

Exact exponent vectors of the variable images.

PythonRust

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

noether::commalg::MonomialMap::images · Direct Rust declaration

pub fn images(&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/MonomialMap@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:198 · validation/unit/commalg/test_polynomial_homomorphism.py:30 · validation/unit/groups/test_acceptance_workload.py:90 · validation/unit/groups/test_acceptance_workload.py:95 · validation/unit/groups/test_permutation_group_facade.py:49 · validation/unit/groups/test_permutation_group_facade.py:54

commalg.MonomialMap.to_json

Serialize this MonomialMap to its canonical typed JSON representation.

Detailed contract

Serialize the exact exponent matrix as canonical typed JSON.

PythonRust

(self) -> str

noether::commalg::MonomialMap::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/MonomialMap@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