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commalg · PolynomialPrimeLocalElement

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

Worked example

The complete checked program Calculate in a fixed-cap multivariate weighted power-series ring uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Calculate in a fixed-cap multivariate weighted power-series ring."""

from noether import Z
from noether.commalg import PowerSeriesRing, SeriesTruncation


def main() -> None:
    ring = PowerSeriesRing(
        Z,
        ["x", "y", "z"],
        SeriesTruncation.weighted_degree([1, 2, 3], 6),
    )
    x, y, z = ring.generators()
    assert (x**6).is_zero()
    assert not (x**2 * z + y * z).is_zero()
    value = (1 + x + y).pow(4)
    print(f"{value}; coefficient capacity <= {ring.coefficient_capacity_bound()}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/17_weighted_power_series.py.

commalg.PolynomialPrimeLocalElement

One exact fraction in a polynomial prime localization.

class PolynomialPrimeLocalElement

noether::commalg::PolynomialPrimeLocalElement · Direct Rust declaration

pub struct PolynomialPrimeLocalElement<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/PolynomialPrimeLocalElement@1

Executable usage: validation/unit/commalg/test_prime_localization.py:31 · validation/unit/commalg/test_prime_localization.py:8

commalg.PolynomialPrimeLocalElement.__add__

Add another compatible value to this value.

Detailed contract

Returns the argument unchanged.

(self, other: PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]) -> PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]

noether::commalg::PolynomialPrimeLocalElement::add · Direct Rust declaration

fn add(self, rhs: Self) -> Self::Output
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/PolynomialPrimeLocalElement@1

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

commalg.PolynomialPrimeLocalElement.__mul__

Multiply this value by another compatible value.

(self, other: PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]) -> PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]

noether::commalg::PolynomialPrimeLocalElement::mul · Direct Rust declaration

fn mul(self, rhs: Self) -> Self::Output
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/PolynomialPrimeLocalElement@1

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

commalg.PolynomialPrimeLocalElement.__neg__

Return the additive inverse of this value.

(self) -> PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]

noether::commalg::PolynomialPrimeLocalElement::neg · Direct Rust declaration

fn neg(self) -> Self::Output
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/PolynomialPrimeLocalElement@1

commalg.PolynomialPrimeLocalElement.__sub__

Subtract another compatible value from this value.

(self, other: PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]) -> PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]

noether::commalg::PolynomialPrimeLocalElement::sub · Direct Rust declaration

fn sub(self, rhs: Self) -> Self::Output
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/PolynomialPrimeLocalElement@1

Executable usage: validation/unit/infra/test_discovery.py:212

commalg.PolynomialPrimeLocalElement.__truediv__

Divide this value by another compatible value.

(self, other: PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]) -> PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]

noether::commalg::PolynomialPrimeLocalElement · 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/PolynomialPrimeLocalElement@1

commalg.PolynomialPrimeLocalElement.denominator

Return the denominator recorded by this PolynomialPrimeLocalElement.

Detailed contract

Canonical denominator outside the localization prime.

(self) -> Polynomial[_ProjectedIdealFieldContext]

noether::commalg::PolynomialPrimeLocalElement::denominator · Direct Rust declaration

pub fn denominator(&self) -> &Polynomial<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/PolynomialPrimeLocalElement@1

Executable usage: validation/unit/commalg/test_curve_facade.py:51 · validation/unit/commalg/test_curve_facade.py:52 · validation/unit/commalg/test_curve_facade.py:93 · validation/unit/commalg/test_curve_facade.py:96 · validation/unit/commalg/test_local_algebra.py:30 · validation/unit/commalg/test_local_algebra.py:46

commalg.PolynomialPrimeLocalElement.from_json

Deserialize and revalidate a PolynomialPrimeLocalElement from typed JSON.

Detailed contract

Deserialize and revalidate a rational prime-local fraction. Deserialize and revalidate a finite-field prime-local fraction.

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

noether::commalg::PolynomialPrimeLocalElement::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/PolynomialPrimeLocalElement@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.PolynomialPrimeLocalElement.inverse

Compute inverse for this PolynomialPrimeLocalElement.

Detailed contract

Invert this value when its numerator lies outside the prime.

(self) -> PolynomialPrimeLocalElement[_ProjectedIdealFieldContext]

noether::commalg::PolynomialPrimeLocalElement::inverse · Direct Rust declaration

pub fn inverse(&self) -> Result<Self, PrimeLocalizationError>
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/PolynomialPrimeLocalElement@1

Executable usage: examples/distributed/f2_linear_algebra.py:37 · examples/distributed/f2_linear_algebra.py:39 · validation/unit/commalg/test_context_tower_workload.py:121 · validation/unit/commalg/test_context_tower_workload.py:122 · validation/unit/commalg/test_context_tower_workload.py:75 · validation/unit/commalg/test_context_tower_workload.py:77

commalg.PolynomialPrimeLocalElement.is_unit

Test whether this PolynomialPrimeLocalElement is unit.

Detailed contract

Decide whether this fraction is a unit.

(self) -> bool

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

pub fn is_unit(&self) -> Result<bool, PrimeLocalizationError>
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/PolynomialPrimeLocalElement@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.PolynomialPrimeLocalElement.is_zero

Test whether this PolynomialPrimeLocalElement is zero.

Detailed contract

Whether this fraction is zero.

(self) -> bool

noether::commalg::PolynomialPrimeLocalElement::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/PolynomialPrimeLocalElement@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.PolynomialPrimeLocalElement.numerator

Return the numerator recorded by this PolynomialPrimeLocalElement.

Detailed contract

Canonical numerator.

(self) -> Polynomial[_ProjectedIdealFieldContext]

noether::commalg::PolynomialPrimeLocalElement::numerator · Direct Rust declaration

pub fn numerator(&self) -> &Polynomial<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/PolynomialPrimeLocalElement@1

Executable usage: validation/unit/commalg/test_curve_facade.py:51 · validation/unit/commalg/test_curve_facade.py:52 · validation/unit/commalg/test_curve_facade.py:93 · validation/unit/commalg/test_curve_facade.py:95 · validation/unit/commalg/test_local_algebra.py:29 · validation/unit/commalg/test_monomial_ideal_facade.py:86

commalg.PolynomialPrimeLocalElement.ring

Return the ring recorded by this PolynomialPrimeLocalElement.

Detailed contract

Exact prime-local context.

(self) -> PolynomialPrimeLocalization[_ProjectedIdealFieldContext]

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

pub fn ring(&self) -> &PolynomialPrimeLocalization<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/PolynomialPrimeLocalElement@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.PolynomialPrimeLocalElement.to_json

Serialize this PolynomialPrimeLocalElement to its canonical typed JSON representation.

Detailed contract

Serialize this canonical prime-local fraction.

(self) -> str

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

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