ideal.ring.md

Polynomial Ring and Ideal Lattice in JavaScript

Given the mathematical concepts of Polynomial Ring and Ideal Lattice, we can implement them in JavaScript.

Definitions:

1. Polynomial Ring: A polynomial ring is a set of polynomials in one or more indeterminates with coefficients in another ring,

2. often a field. The polynomial ring, $$( K[X] )$$, in ( X ) over a field ( K ) can be defined as the set of expressions,

3. called polynomials in ( X ), of the form: $$[ p = p_m X^m + p_{m-1} X^{m-1} + ... + p_1 X + p_0 ]$$ where ( p_0, p_1, …, p_m ) are elements of ( K ) and $$( X, X^2, … )$$ are symbols considered as "powers" of $$( X )$$.

4. Ideal Lattice: Ideal lattices are lattices corresponding to ideals in rings of the form $$( \mathbb{Z}[x]/\langle f \rangle ) $$ for some irreducible polynomial ( f ) of degree ( n ).

JavaScript Implementation:

class IdealLattice {
  constructor(polynomialDegree, irreduciblePolynomial) {
    this.n = polynomialDegree;
    this.f = irreduciblePolynomial;
  }
}

class PolynomialRing extends IdealLattice {
  constructor(coefficients) {
    super(coefficients.length - 1, PolynomialRing.polynomialToString(coefficients));
    this.coefficients = coefficients;
  }

  static polynomialToString(coefficients) {
    return coefficients.map((coeff, index) => `${coeff}x^${index}`).join(' + ');
  }

  evaluate(xValue) {
    return this.coefficients.reduce((acc, coeff, index) => acc + coeff * Math.pow(xValue, index), 0);
  }
}

// Example usage:
const polynomialRing = new PolynomialRing([1, 0, -3, 4]); // Represents 1 + 0x - 3x^2 + 4x^3
console.log(polynomialRing);
console.log(`Value at x=2: ${polynomialRing.evaluate(2)}`);