Polynomial Calculator

A lightweight Computer Algebra System for polynomial computations over finite fields

Features

• Polynomial operations: +, -, *, /, %, gcd
• Supports polynomials over field of characteristic > 0 (A field of characteristic $$p > 0$$ is one where arithmetic behaves like modulo $$p$$, meaning $$p$$ copies of 1 add up to 0).
• Supports multi-variable polynomials
• Numerical solutions for single-variable polynomials
• Numerical solutions for systems of polynomials
• Support for constructing finite fields with polynomials
• Can find Gröbner basis for polynomial Ideals

Command Line Interface

PolynomialCalculator provides a powerful command-line interface with the following commands:

• finite_field: Create finite fields and display Zech logarithm information
• random_monic: Generate random monic polynomials over finite fields
• find_irreducible: Find irreducible polynomials of specified degree
• gcd: Compute greatest common divisor of polynomials
• solve: Solve univariate polynomial equations over rationals
• groebner: Compute Gröbner basis for polynomial ideals

Global options:

• --numeric-output {float,rational} to control numeric display (default: float)
• Convenience flags: --float, --rational
• Logging: --verbose for debug logs, --quiet to show errors only
• Machine-readable: --json to emit JSON for solve and solve-system (also supported for groebner)

Use polycalc --help or polycalc <command> --help for detailed usage information.

Alternative Usage: If you haven't installed the package, you can substitute:

• python polycalc.py (using the wrapper script)
• python -m polynomials.cli (running as a module)

Usage Examples

Create a finite field GF(2^3):

polycalc finite_field -p 2 3

Created GF(2^3) with irreducible polynomial: x^3 + x + 1
Primitive element table size: 7 elements

Generate a random monic polynomial of degree 4 over F_5:

polycalc random_monic -p 5 4

Random monic polynomial over F_5 of degree 4: x^4 + 3x^3 + x^2 + 2x + 2

Find an irreducible polynomial of degree 3 over F_3:

polycalc find_irreducible -p 3 3

Irreducible polynomial over F_3 of degree 3: x^3 + 2x^2 + x + 1

Compute the GCD of two polynomials over F_2:

polycalc gcd "x^3+x+1" "x^2+1" -p 2

gcd(x^3 + x + 1, x^2 + 1) = 1

Solve a univariate polynomial equation:

polycalc solve "x^2-2" x

Solutions to x^2-2 = 0:
  1.4142135623730951
  -1.4142135623730951

Compute a Groebner basis for a system of polynomials:

polycalc groebner "x^2+y^2-1" "x-y"

Groebner basis:
  x - y
  2y^2 - 1

You can choose the monomial order with --order {lex,grlex,grevlex}:

polycalc groebner "x^2+y^2-1" "x-y" --order grevlex

Control numeric output formatting

Default is float mode. Use --rational to prefer exact-looking output when possible.

Environment variable:

• Set POLYCALC_NUMERIC_OUTPUT to float or rational to control the default mode.

polycalc --rational solve-system "x-1" "y-2"

1 solutions:
  [ x = 1, y = 2 ]

polycalc --float solve-system "x-1" "y-2"

1 solutions:
  [ x = 1.0, y = 2.0 ]

JSON output

Add --json to solve, solve-system, or groebner for structured output.

Examples:

polycalc --json solve "x^2-2" x

Outputs:

{ "command": "solve", "poly": "x^2-2", "var": "x", "solutions": [1.4142135623730951, -1.4142135623730951], "count": 2, "status": "ok" }

polycalc --json solve-system "x-1" "y-2"

Outputs:

{ "command": "solve-system", "polys": ["x-1", "y-2"], "status": "ok", "solutions": [{"x": 1.0, "y": 2.0}], "count": 1 }

On errors, JSON mode returns:

{ "status": "error", "error": "<message>" }

Groebner example:

polycalc --json groebner "x^2+y^2-1" "x-y" --order grevlex

Outputs:

{ "command": "groebner", "polys": ["x^2+y^2-1","x-y"], "order": "grevlex", "status": "ok", "basis": ["x - y", "2y^2 - 1"], "count": 2 }

Logging and diagnostics

• CLI logging:
  • --verbose enables debug-level logs for core polynomial and algebra operations.
  • --quiet suppresses non-error output (only errors are shown).
• Core debug tracing:
  • Set environment variable POLYCALC_DEBUG=1 to enable additional debug traces in some core algorithms (e.g., division algorithm and gcd).
  • This is off by default and only emits when combined with a suitable logging level.

Near-term goals:

• implement faster gcd algorithm
• implement lookup tables for primitive field elements
• implement 'factor' algorithm to factor over polynomials over finite fields
• option to plot single-variable polynomials
• create a GUI with Python

Longer-term goals

• web GUI
• Mobile app

Installation

Option 1: Install as a package (Recommended)

git clone https://github.com/mike006322/PolynomialCalculator.git
cd PolynomialCalculator
pip install -e .

After installation, you can use the polycalc command directly:

polycalc solve "x^2-2" x

Option 2: Use the wrapper script

If you prefer not to install the package, you can use the provided wrapper script:

git clone https://github.com/mike006322/PolynomialCalculator.git
cd PolynomialCalculator
python polycalc.py solve "x^2-2" x

Option 3: Run as a Python module

Optional features (algebra extras)

Some features depend on optional scientific libraries. Install with extras:

pip install "PolynomialCalculator[algebra]"

This pulls in numpy and scipy for algebra/finite field operations.

python -m polynomials.cli solve "x^2-2" x

Contributing

Pull requests are welcome! For major changes, please open an issue first to discuss what you’d like to change.

Please make sure to update tests as appropriate.

Developer tooling:

• Lint/format: ruff and black (configured via pyproject.toml)
• Pre-commit hooks: install with pip install pre-commit then pre-commit install
• CI: GitHub Actions runs tests on Python 3.10–3.12 across Windows/Linux/macOS

Performance benchmarking:

• See docs/SPEED_GUIDE.md for how we document speed changes.
• Benchmarks live in benchmarks/ and use pytest-benchmark.

Similar projects:

• SAGE

• Sympy

• CoCa written in C++ and CoCa language

• jasymchat - Computer Algebra System written in Java (won't run on phone, seems defunct)