commalg · RealRootIsolation

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

Worked example

The complete checked program Factor an integer and construct a cyclotomic polynomial uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Factor an integer and construct a cyclotomic polynomial."""

from noether.numthy import cyclotomic_polynomial, factor, is_prime


def main() -> None:
    integer = 2**20 * 3**7 * 101
    factorization = factor(integer)
    assert factorization.verify(integer)
    assert all(is_prime(prime) for prime, _ in factorization.factors)
    phi = cyclotomic_polynomial(12)
    assert phi.terms() == [([4], 1), ([2], -1), ([0], 1)]
    print(f"{integer} = {factorization}; Phi_12 = {phi}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/07_integer_factorization.py.

commalg.RealRootIsolation

One exact real root isolated between rational endpoints.

PythonRust

class RealRootIsolation

noether::commalg::RealRootIsolation · Direct Rust declaration

pub struct RealRootIsolation { /* 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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:104 · validation/unit/commalg/test_polynomial_facade.py:18 · validation/unit/commalg/test_polynomial_facade.py:95

commalg.RealRootIsolation.ball

Convert an exact refinement into a rigorous Arb enclosure.

Detailed contract

Return an Arb enclosure after exact rational refinement. Exact Sturm refinement is performed to eight guard bits beyond the Arb working precision. Arb then encloses both rational endpoints; it is an accelerator and representation layer, not the source of the isolation proof.

PythonRust

(self, precision: int) -> RealBall

noether::commalg::RealRootIsolation::ball · Direct Rust declaration

pub fn ball(&self, precision: u64) -> Result<RealBall, RealRootError>

Contract	Value
Guarantee	rigorous enclosure
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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:102 · validation/corpus/commalg/k3-univariate-certificates.py:109

commalg.RealRootIsolation.ceiling

Least integer not smaller than this root.

Detailed contract

Least integer not smaller than this exact root.

PythonRust

(self) -> int

noether::commalg::RealRootIsolation::ceiling · Direct Rust declaration

pub fn ceiling(&self) -> Result<Integer, RealRootError>

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:106 · validation/unit/commalg/test_polynomial_facade.py:108 · validation/unit/infra/test_slurm_handoff.py:344

commalg.RealRootIsolation.compare

Exact comparison with another isolated root.

Detailed contract

Compare two exact isolated roots, recognizing equal algebraic values.

PythonRust

(self, other: RealRootIsolation) -> int

noether::commalg::RealRootIsolation::compare · Direct Rust declaration

pub fn compare(&self, other: &Self) -> Result<Ordering, RealRootError>

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:109 · validation/backends/graph/invariants.py:67 · validation/bench/_macaulay2.py:5 · validation/bench/groebner_compare.py:457 · validation/bench/sage_exterior_algebra.py:5 · validation/harness/parity.py:20

commalg.RealRootIsolation.compare_rational

Exact comparison with a rational, returned as -1, 0, or 1.

Detailed contract

Compare this exact root with a rational number.

PythonRust

(self, value: int | _Fraction) -> int

noether::commalg::RealRootIsolation::compare_rational · Direct Rust declaration

pub fn compare_rational( &self, value: &Rational, ) -> Result<Ordering, RealRootError>

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:110

commalg.RealRootIsolation.defining_factor

Monic square-free defining factor carrying this root.

Detailed contract

Monic square-free defining factor carrying the isolated root.

PythonRust

(self) -> Polynomial[RationalField]

noether::commalg::RealRootIsolation::defining_factor · Direct Rust declaration

pub fn defining_factor(&self) -> &Polynomial<RationalField>

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:97

commalg.RealRootIsolation.floor

Greatest integer not exceeding this root.

Detailed contract

Greatest integer not exceeding this exact root.

PythonRust

(self) -> int

noether::commalg::RealRootIsolation::floor · Direct Rust declaration

pub fn floor(&self) -> Result<Integer, RealRootError>

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:106 · validation/unit/commalg/test_polynomial_facade.py:108

commalg.RealRootIsolation.from_json

Deserialize and revalidate a RealRootIsolation from typed JSON.

Detailed contract

Deserialize and exactly verify a typed certificate.

PythonRust

(cls, value: str) -> RealRootIsolation

noether::commalg::RealRootIsolation::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/RealRootIsolation@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.RealRootIsolation.lower

Strict lower rational endpoint.

PythonRust

(self) -> _Fraction

noether::commalg::RealRootIsolation::lower · Direct Rust declaration

pub fn lower(&self) -> &Rational

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:101 · validation/unit/commalg/test_polynomial_facade.py:98 · validation/unit/commalg/test_polynomial_facade.py:99 · validation/unit/commalg/test_surface_facade.py:72 · validation/unit/commalg/test_surface_facade.py:74 · validation/unit/commalg/test_surface_facade.py:76

commalg.RealRootIsolation.multiplicity

Exact root multiplicity.

Detailed contract

Multiplicity in the stored polynomial.

PythonRust

(self) -> int

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

pub fn multiplicity(&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/RealRootIsolation@1

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

commalg.RealRootIsolation.polynomial

Exact rational polynomial whose root is isolated.

Detailed contract

Original rational polynomial whose root is isolated.

PythonRust

(self) -> Polynomial[RationalField]

noether::commalg::RealRootIsolation::polynomial · Direct Rust declaration

pub fn polynomial(&self) -> &Polynomial<RationalField>

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/RealRootIsolation@1

Executable usage: examples/07_integer_factorization.py:3 · examples/14_weighted_polynomials.py:13 · examples/14_weighted_polynomials.py:15 · examples/14_weighted_polynomials.py:3 · examples/22_matroids_and_tutte.py:3 · examples/27_weight_characters.py:3

commalg.RealRootIsolation.refine

Refine to an interval of width at most 2^-bits.

Detailed contract

Refine the exact interval to width at most 2^-bits.

PythonRust

(self, bits: int) -> RealRootIsolation

noether::commalg::RealRootIsolation::refine · Direct Rust declaration

pub fn refine(&self, bits: u64) -> Result<Self, RealRootError>

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:100 · validation/corpus/commalg/k3-univariate-certificates.py:108

commalg.RealRootIsolation.sign

Exact sign, -1, 0, or 1.

Detailed contract

Sign of this exact root: -1, 0, or 1.

PythonRust

(self) -> int

noether::commalg::RealRootIsolation::sign · Direct Rust declaration

pub fn sign(&self) -> Result<i8, RealRootError>

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:105 · validation/unit/commalg/test_polynomial_facade.py:107 · validation/parity/commalg/test_intersection_theory_sage_parity.py:146 · validation/parity/commalg/test_intersection_theory_sage_parity.py:149 · validation/parity/commalg/test_intersection_theory_sage_parity.py:151 · validation/corpus/commalg/k3-univariate-certificates.py:107

commalg.RealRootIsolation.to_json

Serialize this RealRootIsolation to its canonical typed JSON representation.

Detailed contract

Serialize this exact certificate as typed JSON.

PythonRust

(self) -> str

noether::commalg::RealRootIsolation::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/RealRootIsolation@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.RealRootIsolation.upper

Strict upper rational endpoint.

PythonRust

(self) -> _Fraction

noether::commalg::RealRootIsolation::upper · Direct Rust declaration

pub fn upper(&self) -> &Rational

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/RealRootIsolation@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:101 · validation/unit/commalg/test_polynomial_facade.py:98 · validation/unit/commalg/test_polynomial_facade.py:99 · validation/unit/commalg/test_surface_facade.py:73 · validation/unit/commalg/test_surface_facade.py:75 · validation/unit/commalg/test_surface_facade.py:76

commalg.RealRootIsolation.verify

Verify the exact Sturm and multiplicity certificate.

Detailed contract

Verify the Sturm root count and exact multiplicity certificate.

PythonRust

(self) -> None

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

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

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/RealRootIsolation@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