Database of indecomposable elements over number fields

This is a (work-in-progress) database of additively indecomposable totally positive integral elements over number fields.

For a given a number field $K$, an element $\alpha \in K$ is totally positive if $\sigma(\alpha) > 0$ for all real embeddings $\sigma$ of $K$. We say that a totally positive integral element $\alpha \in \mathcal{O}_K$ is (additively) indecomposable if it cannot be written as the sum of two other totally positive integral elements in $\mathcal{O}_K$.

Given a number field $K$, it's known that there are only finitely many indecomposable elements up to multiplicatiion by totally positive units. There is however no known general classification of all such indecomposable elements (other than some particular families of number fields, e.g. real quadratic fields. simplest cubic fields, or certain families of biquadratic fields).

Sage code to enumerate all additively indecomposable elements (modulo totally positive units) is provided in the scripts folder, tables of data is given in the data folder, and various plots of the data is given in the plots folder.

Any contributions to either the code or data are very welcome!

Usage

After installing Sage (see Development section below), you can compute indecomposables for totally real number fields using the provided shell script.

Basic Usage

Compute indecomposables for degree 3 fields with discriminants between 1 and 100:

./compute_indecomposables.sh 3 1 100

This will:

• Process all degree 3 fields from totally_real_fields/deg3.txt
• Filter fields with discriminants in the range [1, 100]
• Output results to results_deg3.txt
• Show basic progress information

Command Syntax

./compute_indecomposables.sh DEGREE DISC_MIN DISC_MAX [options]

Required Arguments:

• DEGREE: Degree of number fields to process (3, 4, 5, etc.)
• DISC_MIN: Minimum discriminant value
• DISC_MAX: Maximum discriminant value (required)

Options:

• -v, --verbose: Enable verbose console output (shows detailed progress)
• -d, --debug: Enable debug console logging
• --num-threads N: Set total thread count for parallel processing (default: 1)
• --thread-id I: Set thread ID for distributed computation (default: 0)
• -h, --help: Show help message

Examples

# Degree 3 fields, discriminants 1-100, basic mode
./compute_indecomposables.sh 3 1 100

# Degree 4 fields, discriminants 1-500, with verbose output
./compute_indecomposables.sh 4 1 500 --verbose

# Large computation with multiple threads
./compute_indecomposables.sh 5 1 10000 --num-threads 4

# Distributed computation (process part of a larger range)
./compute_indecomposables.sh 3 1 10000 --num-threads 4 --thread-id 0

# With debug logging
./compute_indecomposables.sh 3 1 100 --debug

Parallel Processing

The script supports distributed computation across multiple processes. Use run_parallel.sh to automatically split the work:

./run_parallel.sh DEGREE DISC_MIN DISC_MAX NUM_PROCESSES

Development

Setup

This project requires Sage for number field computations.

Project Structure

indecomposables/
├── scripts/           # Core computation code
│   ├── main.py       # NumberFieldData class with optimized algorithm selection
│   ├── real_quadratic.py      # Dress-Scharlau algorithm for quadratic fields
│   ├── simplest_cubic.py      # Kala-Tinková classification for cubic fields
│   ├── demo_real_quadratic.py # Performance demo for quadratic fields
│   └── demo_simplest_cubic.py # Performance demo for cubic fields
├── data/             # Database files (pipe-delimited, PostgreSQL-compatible)
├── plots/            # Visualization and analysis
├── tests/            # Test suite
│   ├── test_main.py          # Main NumberFieldData tests
│   ├── test_real_quadratic.py # RealQuadraticField tests
│   └── test_simplest_cubic.py # SimplestCubicField tests
└── Makefile          # Development commands

Main API

See scripts/README.md for detailed API documentation of the NumberFieldData class.

Quick example:

from sage.all import QQ, PolynomialRing
from main import NumberFieldData

# Create a number field
R = PolynomialRing(QQ, 'x')
K = QQ.extension(R.gen()**3 - 2, names='a')

# Compute indecomposables
nfd = NumberFieldData(label="3.3.49.1", field=K)
indecomps = nfd.compute_indecomposables()

print(f"Found {len(indecomps)} indecomposables")