Groebner Bases

Topic: polynomial.groebner

Groebner bases are fundamental tools in computational algebraic geometry for working with polynomial ideals, enabling ideal membership testing, polynomial system solving, variable elimination, and geometric theorem proving.

Mathematical Definition

Groebner Basis Definition: A set $G = {g_{1}, \dots, g_{m}}$ is a Groebner basis for ideal $I$ with respect to monomial order $<$ if:

$⟨ LT (g_{1}), \dots, LT (g_{m})⟩ = ⟨ LT (I)⟩$

where $LT$ denotes the leading term and $⟨ \cdot ⟩$ denotes the ideal generated.

S-Polynomial: The S-polynomial of $f$ and $g$ is:

$S (f, g) = \frac{lcm ( LT ( f ) , LT ( g ))}{LT ( f )} \cdot f - \frac{lcm ( LT ( f ) , LT ( g ))}{LT ( g )} \cdot g$

Buchberger's Criterion: $G$ is a Groebner basis if and only if for all pairs $g_{i}, g_{j} \in G$ :

$S (g_{i}, g_{j}) G_{+} 0$

(i.e., $S (g_{i}, g_{j})$ reduces to 0 modulo $G$ )

Monomial Orders:

Lex: $x^{α} > x^{β} ⟺$ first non-zero entry of $α - β$ is positive
Grlex: Total degree first, then lex
Grevlex: Total degree first, then reverse lex from right

Groebner bases are a fundamental tool in computational algebraic geometry for working with polynomial ideals.

Overview

A Groebner basis is a special generating set for a polynomial ideal that has many useful computational properties:

Ideal Membership Testing: Determine if a polynomial belongs to an ideal
Polynomial System Solving: Find common solutions to systems of polynomial equations
Variable Elimination: Eliminate variables from polynomial systems
Geometric Theorem Proving: Prove geometric theorems algebraically

Monomial Orders

The choice of monomial order affects the structure of the Groebner basis:

Order	Description	Use Case
`Lex`	Lexicographic	Variable elimination
`Grlex`	Graded lexicographic	Balanced computation
`Grevlex`	Graded reverse lexicographic	Efficient computation

Lexicographic Order (Lex)

Compares exponents from left to right:

$x^{2} y > x y^{2}$ (2 > 1 in first position)
$x y^{3} > x z^{5}$ (y > z in second position)

Best for: Variable elimination, solving systems

Graded Lexicographic (Grlex)

Compares total degree first, then lexicographic:

$x y^{2} > x^{2}$ (degree 3 > 2)
$x^{2} > x y$ (same degree, $x^{2} > x y$ lexicographically)

Best for: Balanced trade-off between structure and efficiency

Graded Reverse Lexicographic (Grevlex)

Compares total degree first, then reverse lexicographic from right:

$x y^{2} > x^{2} y$ (same degree, compare from right)

Best for: Efficient computation (often produces smaller bases)

Buchberger's Algorithm

The classic algorithm for computing Groebner bases:

Algorithm Steps

Initialize: Start with the input polynomials
S-pairs: For each pair of polynomials, compute the S-polynomial
Reduce: Reduce each S-polynomial by the current basis
Add: If reduction is non-zero, add to basis
Repeat: Continue until no new polynomials are added

Only z (elimination of x and y)
y and z (elimination of x)
x, y, and z

Examples

Basic Groebner Basis Computation

Compute Groebner basis for a polynomial ideal

Rust

#![allow(unused)]
fn main() {
use mathhook_core::core::polynomial::groebner::{GroebnerBasis, MonomialOrder};
use mathhook_core::{expr, symbol};

let x = symbol!(x);
let y = symbol!(y);

// Define polynomials: f1 = x - y, f2 = y^2 - 1
let f1 = expr!(x - y);
let f2 = expr!((y ^ 2) - 1);

// Create Groebner basis
let mut gb = GroebnerBasis::new(
    vec![f1, f2],
    vec![x.clone(), y.clone()],
    MonomialOrder::Lex
);

// Compute the basis
gb.compute();

println!("Basis has {} polynomials", gb.basis.len());

}

Python

from mathhook import expr, symbol
from mathhook.polynomial.groebner import GroebnerBasis, MonomialOrder

x = symbol('x')
y = symbol('y')

# Define polynomials: f1 = x - y, f2 = y^2 - 1
f1 = expr('x - y')
f2 = expr('y^2 - 1')

# Create Groebner basis
gb = GroebnerBasis(
    [f1, f2],
    [x, y],
    MonomialOrder.Lex
)

# Compute the basis
gb.compute()

print(f"Basis has {len(gb.basis)} polynomials")

JavaScript

const { expr, symbol } = require('mathhook');
const { GroebnerBasis, MonomialOrder } = require('mathhook/polynomial/groebner');

const x = symbol('x');
const y = symbol('y');

// Define polynomials: f1 = x - y, f2 = y^2 - 1
const f1 = expr('x - y');
const f2 = expr('y^2 - 1');

// Create Groebner basis
const gb = new GroebnerBasis(
    [f1, f2],
    [x, y],
    MonomialOrder.Lex
);

// Compute the basis
gb.compute();

console.log(`Basis has ${gb.basis.length} polynomials`);

Sparse Polynomial Representation

Work with sparse polynomials for efficiency

Rust

#![allow(unused)]
fn main() {
use mathhook_core::core::polynomial::groebner::{Monomial, expression_to_sparse_polynomial};
use mathhook_core::{expr, symbol};

// Create a monomial x^2 * y (exponents [2, 1])
let mono = Monomial::new(vec![2, 1]);
assert_eq!(mono.degree(), 3);

// Convert expression to sparse polynomial
let x = symbol!(x);
let y = symbol!(y);
let poly = expr!((x ^ 2) + y);

let vars = vec![x, y];
let sparse = expression_to_sparse_polynomial(&poly, &vars);

}

Python

from mathhook import expr, symbol
from mathhook.polynomial.groebner import Monomial, expression_to_sparse_polynomial

# Create a monomial x^2 * y (exponents [2, 1])
mono = Monomial([2, 1])
assert mono.degree() == 3

# Convert expression to sparse polynomial
x = symbol('x')
y = symbol('y')
poly = expr('x^2 + y')

vars = [x, y]
sparse = expression_to_sparse_polynomial(poly, vars)

JavaScript

const { expr, symbol } = require('mathhook');
const { Monomial, expressionToSparsePolynomial } = require('mathhook/polynomial/groebner');

// Create a monomial x^2 * y (exponents [2, 1])
const mono = new Monomial([2, 1]);
assert(mono.degree() === 3);

// Convert expression to sparse polynomial
const x = symbol('x');
const y = symbol('y');
const poly = expr('x^2 + y');

const vars = [x, y];
const sparse = expressionToSparsePolynomial(poly, vars);

Polynomial Reduction

Reduce polynomial by a set of polynomials (division algorithm)

Rust

#![allow(unused)]
fn main() {
use mathhook_core::core::polynomial::groebner::{
    poly_reduce,
    poly_reduce_completely
};

// Single-step reduction
let reduced = poly_reduce(&poly, &basis, &order);

// Complete reduction (until no further reduction possible)
let fully_reduced = poly_reduce_completely(&poly, &basis, &order);

}

Python

from mathhook.polynomial.groebner import poly_reduce, poly_reduce_completely

# Single-step reduction
reduced = poly_reduce(poly, basis, order)

# Complete reduction (until no further reduction possible)
fully_reduced = poly_reduce_completely(poly, basis, order)

JavaScript

const { polyReduce, polyReduceCompletely } = require('mathhook/polynomial/groebner');

// Single-step reduction
const reduced = polyReduce(poly, basis, order);

// Complete reduction (until no further reduction possible)
const fullyReduced = polyReduceCompletely(poly, basis, order);

Bidirectional Expression Conversion

Convert between Expression and sparse polynomial representation

Rust

#![allow(unused)]
fn main() {
use mathhook_core::core::polynomial::groebner::{
    expression_to_sparse_polynomial,
    sparse_polynomial_to_expression
};
use mathhook_core::{expr, symbol};

let x = symbol!(x);
let y = symbol!(y);
let vars = vec![x.clone(), y.clone()];

// Expression to sparse
let expr = expr!((x ^ 2) + y);
let sparse = expression_to_sparse_polynomial(&expr, &vars).unwrap();

// Sparse back to expression
let back = sparse_polynomial_to_expression(&sparse, &vars);

}

Python

from mathhook import expr, symbol
from mathhook.polynomial.groebner import (
    expression_to_sparse_polynomial,
    sparse_polynomial_to_expression
)

x = symbol('x')
y = symbol('y')
vars = [x, y]

# Expression to sparse
e = expr('x^2 + y')
sparse = expression_to_sparse_polynomial(e, vars)

# Sparse back to expression
back = sparse_polynomial_to_expression(sparse, vars)

JavaScript

const { expr, symbol } = require('mathhook');
const {
    expressionToSparsePolynomial,
    sparsePolynomialToExpression
} = require('mathhook/polynomial/groebner');

const x = symbol('x');
const y = symbol('y');
const vars = [x, y];

// Expression to sparse
const e = expr('x^2 + y');
const sparse = expressionToSparsePolynomial(e, vars);

// Sparse back to expression
const back = sparsePolynomialToExpression(sparse, vars);

Performance

Time Complexity: Doubly exponential worst case, practical for small systems

API Reference

Rust: mathhook_core::polynomial::groebner
Python: mathhook.polynomial.groebner
JavaScript: mathhook.polynomial.groebner

MathHook KB Documentation

Groebner Bases

Mathematical Definition

Overview

Monomial Orders

Lexicographic Order (Lex)

Graded Lexicographic (Grlex)

Graded Reverse Lexicographic (Grevlex)

Buchberger's Algorithm

Algorithm Steps

Applications

Solving Polynomial Systems

Ideal Membership

Elimination

Examples

Basic Groebner Basis Computation

Sparse Polynomial Representation

Polynomial Reduction

Bidirectional Expression Conversion

Performance

API Reference

See Also

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MathHook KB Documentation