Struct GroebnerBasis 

pub struct GroebnerBasis {
    pub basis: Vec<Expression>,
    pub variables: Vec<Symbol>,
    pub ordering: MonomialOrder,
    pub is_reduced: bool,
}

Represents a Gröbner basis for a polynomial ideal

A Gröbner basis is a special generating set for a polynomial ideal that has useful computational properties, analogous to row echelon form for matrices or GCD for integers.

Mathematical Background

For an ideal I = <f1, f2, …, fn> in k[x1, …, xm], a Gröbner basis is a finite subset G of I such that:

1. G generates I (every element of I is a polynomial combination of G)
2. The leading terms of G generate the ideal of leading terms of I

Applications

• Ideal membership testing: Check if f ∈ I
• Solving systems of polynomial equations
• Computing ideal operations (intersection, quotient, elimination)
• Implicitization in algebraic geometry
• Computational commutative algebra

Fields

basis: Vec<Expression>

The basis polynomials

variables: Vec<Symbol>

Variables in the polynomial ring

ordering: MonomialOrder

Monomial ordering used for computation

is_reduced: bool

Whether the basis is reduced

Implementations

impl GroebnerBasis

pub fn new( polynomials: Vec<Expression>, variables: Vec<Symbol>, ordering: MonomialOrder, ) -> Self

Create a new Gröbner basis from polynomials

Arguments

• polynomials - Initial generating set for the ideal
• variables - Variables in the polynomial ring
• ordering - Monomial ordering to use

Examples

use mathhook_core::{symbol, expr, Expression};
use mathhook_core::algebra::groebner::{GroebnerBasis, MonomialOrder};

let x = symbol!(x);
let y = symbol!(y);
let f1 = Expression::add(vec![Expression::pow(x.clone().into(), Expression::integer(2)), Expression::pow(y.clone().into(), Expression::integer(2)), Expression::integer(-1)]);
let f2 = Expression::add(vec![x.clone().into(), Expression::mul(vec![Expression::integer(-1), y.clone().into()])]);
let gb = GroebnerBasis::new(
    vec![f1, f2],
    vec![x, y],
    MonomialOrder::Lex
);

pub fn compute(&mut self)

Compute the Gröbner basis using Buchberger’s algorithm

Transforms the initial generators into a Gröbner basis by computing S-polynomials and adding non-zero remainders to the basis.

Examples

use mathhook_core::{symbol, expr};
use mathhook_core::algebra::groebner::{GroebnerBasis, MonomialOrder};

let x = symbol!(x);
let y = symbol!(y);
let f1 = Expression::add(vec![Expression::pow(x.clone().into(), Expression::integer(2)), Expression::pow(y.clone().into(), Expression::integer(2)), Expression::integer(-1)]);
let f2 = Expression::add(vec![x.clone().into(), Expression::mul(vec![Expression::integer(-1), y.clone().into()])]);
let mut gb = GroebnerBasis::new(
    vec![f1, f2],
    vec![x, y],
    MonomialOrder::Lex
);
gb.compute();

pub fn compute_with_result(&mut self) -> MathResult<()>

Compute the Gröbner basis with explicit error handling

Returns Ok(()) on success or Err(MathError) if computation times out or exceeds iteration limit.

Examples

use mathhook_core::{symbol, expr};
use mathhook_core::algebra::groebner::{GroebnerBasis, MonomialOrder};

let x = symbol!(x);
let y = symbol!(y);
let f1 = Expression::add(vec![Expression::pow(x.clone().into(), Expression::integer(2)), Expression::pow(y.clone().into(), Expression::integer(2)), Expression::integer(-1)]);
let f2 = Expression::add(vec![x.clone().into(), Expression::mul(vec![Expression::integer(-1), y.clone().into()])]);
let mut gb = GroebnerBasis::new(
    vec![f1, f2],
    vec![x, y],
    MonomialOrder::Lex
);
if gb.compute_with_result().is_ok() {
    // Computation succeeded
} else {
    // Computation timed out or exceeded iteration limit
}

pub fn reduce(&mut self)

Reduce the Gröbner basis to minimal form

A reduced Gröbner basis has:

1. Leading coefficients are 1 (monic)
2. No monomial of any basis element is divisible by the leading term of another basis element

pub fn contains(&self, poly: &Expression) -> bool

Test if a polynomial is in the ideal generated by this basis

Arguments

• poly - Polynomial to test for membership

Returns

Returns true if the polynomial reduces to zero modulo the basis

Examples

use mathhook_core::{symbol, expr};
use mathhook_core::algebra::groebner::{GroebnerBasis, MonomialOrder};

let x = symbol!(x);
let y = symbol!(y);
let f1 = expr!(x - y);
let f2 = Expression::add(vec![Expression::pow(y.clone().into(), Expression::integer(2)), Expression::integer(-1)]);
let mut gb = GroebnerBasis::new(
    vec![f1, f2],
    vec![x.clone(), y.clone()],
    MonomialOrder::Lex
);
gb.compute();

let test = Expression::add(vec![Expression::pow(x.clone().into(), Expression::integer(2)), Expression::integer(-1)]);
assert!(gb.contains(&test));

pub fn get_variables(&self) -> Vec<Symbol>

Get all variables that appear in the basis

Trait Implementations

impl Clone for GroebnerBasis

fn clone(&self) -> GroebnerBasis

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for GroebnerBasis

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.