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commalg · TateCyclotomicFactor

4 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.TateCyclotomicFactor

One cyclotomic factor of the Tate-normalized Frobenius polynomial.

class TateCyclotomicFactor

noether::commalg::TateCyclotomicFactor · Direct Rust declaration

pub struct TateCyclotomicFactor { /* 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

commalg.TateCyclotomicFactor.degree

Return the degree recorded by this TateCyclotomicFactor.

Detailed contract

Degree phi(n) of this factor.

(self) -> int

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

pub fn degree(&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

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.TateCyclotomicFactor.index

Return the index recorded by this TateCyclotomicFactor.

Detailed contract

Cyclotomic index n for the factor Phi_n.

(self) -> int

noether::commalg::TateCyclotomicFactor::index · Direct Rust declaration

pub fn index(&self) -> u64
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

Executable usage: examples/38_exact_linear_systems.py:79 · validation/unit/combinatorics/test_kashiwara_crystal.py:34 · validation/unit/combinatorics/test_kashiwara_crystal.py:35 · validation/unit/combinatorics/test_kashiwara_crystal.py:36 · validation/unit/commalg/test_base_change.py:90 · validation/unit/commalg/test_base_change.py:93

commalg.TateCyclotomicFactor.multiplicity

Return the multiplicity recorded by this TateCyclotomicFactor.

Detailed contract

Multiplicity in the normalized Frobenius polynomial.

(self) -> int

noether::commalg::TateCyclotomicFactor::multiplicity · Direct Rust declaration

pub fn multiplicity(&self) -> u64
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

Executable usage: validation/unit/combin/test_acceptance_workload.py:64 · validation/unit/commalg/test_local_algebra.py:65 · validation/unit/commalg/test_local_algebra.py:82 · validation/unit/commalg/test_polynomial_facade.py:126 · validation/unit/commalg/test_polynomial_facade.py:158 · validation/unit/commalg/test_polynomial_facade.py:96