Random Polynomial Experiments

A modular Python framework for generating, analyzing, and visualizing random polynomial ensembles, including real and complex coefficients drawn from customizable distributions.

The system simulates ensembles of random polynomials, computes per-trial statistics (e.g. number of real roots, largest real gap, root angles), and provides flexible tools for summary analysis and visualization.

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Project Structure

random-polynomials/
├── main.py              # Entry point to run experiments and visualize results
├── analysis.py          # Core Experiment & ExperimentResult classes
├── viz.py               # Visualization utilities and dashboard API
├── polygen.py           # Random polynomial generator
├── dist.py              # Unified real and complex distribution library
├── polystat.py          # Target functions and numerical statistics
├── requirements.txt     # Python dependencies
└── README.md            # This file :)

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Installation

1. Clone the repository:

   git clone https://github.com/generic-account/random-polynomials.git
   cd random-polynomials

2. Create a virtual environment:

   python3 -m venv .venv
   source .venv/bin/activate

3. Install dependencies:

   pip install -r requirements.txt

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Basic Usage

1. Run an Experiment

main.py provides an entry point to run and visualize an experiment.

python3 main.py

By default, it will:

• Generate random polynomials of a given degree (e.g., 30)
• Use coefficients sampled from a chosen distribution (e.g., Normal(0,1))
• Run a number of trials (e.g., 500)
• Compute statistics and plot visualizations

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2. Example: Custom Experiment

You can modify main.py or run interactively:

from polygen import Poly
from dist import Normal, ComplexNormal, Rademacher
from analysis import Experiment
from viz import Visualizer

# Define polynomial generator
poly = Poly(degree=30, dist=ComplexNormal(0, 1))

# Run experiment
exp = Experiment(poly, trials=500,
                 outputs=["n_real", "real_gap_max", "roots", "real_roots", "angles"])
results = exp.run(summary="basic")

# Visualize
viz = Visualizer(results)
viz.quick_summary()

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3. Output Example

You’ll see output like:

Running experiment: 100%|███████████████████████| 500/500 [00:00<00:00, 950.13it/s]
Summary statistics (basic):
  n_real: {'mean': 4.6, 'std': 2.1, 'min': 0.0, 'max': 10.0, 'median': 5.0}
  real_gap_max: {'mean': 3.5, 'std': 1.2, 'min': 0.8, 'max': 6.7, 'median': 3.2}

and a dashboard of histograms and scatterplots on a single page.

Visualization Features

A simple visualization API:

viz.show([
    ("hist", "angles"),
    ("hist", "real_gap_max"),
    ("scatter_complex", "roots"),
    ("scatter_real", "real_roots"),
], layout="auto", title="Custom Dashboard")

• Multiple plots appear on one page
• Complex root scatterplots are automatically square
• Experiment parameters (degree, distribution, trials) appear in the title

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Example Distributions

Distribution	Type	Description
Normal(0,1)	Real	Standard Gaussian
Uniform(-1,1)	Real	Uniform over [-1,1]
Rademacher()	Real	±1 with equal probability
Laplace(0,1)	Real	Double-exponential
ComplexNormal(0,1)	Complex	Circularly symmetric complex Gaussian
ComplexUniform(radius=1)	Complex	Uniform in unit disk
ComplexRademacher()	Complex	Uniform on unit circle
ComplexLaplace()	Complex	Laplace-distributed real/imag parts

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