Groebner Bases

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.

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Mathematical Definition

Groebner Basis Definition: A set $G = \{g_1, \ldots, g_m\}$ is a Groebner basis for ideal $I$ with respect to monomial order $<$ if:

\langle \text{LT}(g_1), \ldots, \text{LT}(g_m) \rangle = \langle \text{LT}(I) \rangle

where $\text{LT}$ denotes the leading term and $\langle \cdot \rangle$ denotes the ideal generated.

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

S(f,g) = \frac{\text{lcm}(\text{LT}(f), \text{LT}(g))}{\text{LT}(f)} \cdot f - \frac{\text{lcm}(\text{LT}(f), \text{LT}(g))}{\text{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) \xrightarrow{G}_+ 0

(i.e., $S(g_i, g_j)$ reduces to 0 modulo $G$ )

Monomial Orders: - Lex: $x^\alpha > x^\beta \iff$ first non-zero entry of $\alpha - \beta$ is positive - Grlex: Total degree first, then lex - Grevlex: Total degree first, then reverse lex from right

Code Examples

Basic Groebner Basis Computation

Compute Groebner basis for a polynomial ideal

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());

Sparse Polynomial Representation

Work with sparse polynomials for efficiency

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);

Polynomial Reduction

Reduce polynomial by a set of polynomials (division algorithm)

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);

Bidirectional Expression Conversion

Convert between Expression and sparse polynomial representation

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);

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