Buchberger's Algorithm

Definition

Buchberger's Algorithm is used to compute the Gröbner Basis for an ideal in a polynomial ring, which is essential in solving systems of polynomial equations and performing symbolic algebraic computations. The algorithm iteratively computes the S-polynomial of polynomial pairs and reduces it until a basis for the ideal is found.

Characteristics

• Algorithm Type: Symbolic computation, Gröbner Basis computation.
• Main Operation: Computes the S-polynomial of polynomial pairs and reduces them.
• Data Structures: A vector of polynomials is maintained during the computation.
• Output: A Gröbner Basis for the input polynomials.

Time Complexity

• Average Case: The complexity depends on the degree of the polynomials and the number of terms. The algorithm can have exponential time complexity in the worst case.
• Worst Case: $O(n^3)$ or more, depending on the implementation and the number of polynomials involved.

Space Complexity

• Space Complexity: $O(n)$, where $n$ is the number of polynomials. The space is required to store the polynomials and their intermediate results.

Approach

1. Input: Start with a set of polynomials.
2. Compute S-polynomial: For every pair of polynomials in the set, compute their S-polynomial.
3. Reduce S-polynomial: Divide the S-polynomial by the existing polynomials and check the remainder. If the remainder is non-zero, add it to the set of polynomials.
4. Repeat: Continue this process until no new polynomials are generated, i.e., until the basis converges.
5. Output Gröbner Basis: The set of polynomials that have been reduced is the Gröbner basis for the ideal.

#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

// Polynomial structure
struct Polynomial {
    vector<int> coefficients;
    int degree;

    Polynomial(int degree) : degree(degree) {
        coefficients.resize(degree + 1, 0);
    }

    // Utility function to display a polynomial
    void print() {
        for (int i = degree; i >= 0; --i) {
            if (coefficients[i] != 0) {
                cout << coefficients[i] << "x^" << i;
                if (i > 0) cout << " + ";
            }
        }
        cout << endl;
    }
};

// Function to compute the S-polynomial of two polynomials
Polynomial sPolynomial(const Polynomial &f, const Polynomial &g) {
    int lcmDegree = max(f.degree, g.degree);
    Polynomial result(lcmDegree);

    // Implementing the S-polynomial formula: S(f, g) = (leading term of g / leading term of f) * f - (leading term of f / leading term of g) * g
    int lcm = f.coefficients[f.degree] * g.coefficients[g.degree];
    
    for (int i = 0; i <= f.degree; ++i) {
        result.coefficients[i] = f.coefficients[i] * g.coefficients[g.degree] - g.coefficients[i] * f.coefficients[f.degree];
    }
    
    return result;
}

// Function to perform polynomial division and return the remainder
Polynomial dividePolynomial(const Polynomial &f, const Polynomial &g) {
    Polynomial remainder = f;

    while (remainder.degree >= g.degree) {
        int coeff = remainder.coefficients[remainder.degree] / g.coefficients[g.degree];
        int degree = remainder.degree - g.degree;

        for (int i = 0; i <= g.degree; ++i) {
            remainder.coefficients[i + degree] -= coeff * g.coefficients[i];
        }
        remainder.degree--;
    }

    return remainder;
}

// Buchberger's Algorithm to compute Gröbner Basis
vector<Polynomial> buchberger(vector<Polynomial> &polynomials) {
    vector<Polynomial> basis = polynomials;

    bool changed = true;
    while (changed) {
        changed = false;
        for (size_t i = 0; i < basis.size(); ++i) {
            for (size_t j = i + 1; j < basis.size(); ++j) {
                // Compute the S-polynomial of basis[i] and basis[j]
                Polynomial sp = sPolynomial(basis[i], basis[j]);
                // Perform polynomial division to check if the remainder is zero
                Polynomial remainder = dividePolynomial(sp, basis[i]);

                // If remainder is non-zero, add it to the basis
                if (remainder.degree > 0) {
                    basis.push_back(remainder);
                    changed = true;
                }
            }
        }
    }

    return basis;
}

int main() {
    // Example polynomials
    Polynomial p1(2);  // x^2 + 2x + 1
    p1.coefficients[2] = 1;
    p1.coefficients[1] = 2;
    p1.coefficients[0] = 1;

    Polynomial p2(2);  // x^2 + x + 1
    p2.coefficients[2] = 1;
    p2.coefficients[1] = 1;
    p2.coefficients[0] = 1;

    vector<Polynomial> polynomials = {p1, p2};

    vector<Polynomial> grobnerBasis = buchberger(polynomials);

    // Output Gröbner Basis
    cout << "Gröbner Basis: " << endl;
    for (auto &p : grobnerBasis) {
        p.print();
    }

    return 0;
}

---