/**
 * Exact surd values — the engine's exact-number domain beyond ℚ.
 *
 * A value is `q₀ + Σ qᵢ·√nᵢ` (qᵢ ∈ Rational, nᵢ distinct square-free integers).
 * Internally one Map keyed by square-free radicand (`1n` is the rational part);
 * every stored coefficient is nonzero, so canonical form makes equality
 * STRUCTURAL — which is sound because the √ of distinct square-free integers are
 * ℚ-linearly independent, so `q₀ + Σ qᵢ√nᵢ = 0` iff every qᵢ = 0.
 *
 * Closed under `+ − ×`, and under `÷` for the cases quadratic work produces:
 * division by a rational, or by a single-radical value `a + b√n` (via its
 * conjugate). A denominator with TWO OR MORE distinct radicals needs
 * multiquadratic rationalization — out of scope here, so `inverse()` returns
 * `undefined` (an honest undefined point, never an approximation). `√` of a
 * negative is `undefined` too (complex numbers are a later step).
 *
 * No floating point, no DOM — exact bigint/Rational arithmetic only.
 */
import { Rational } from "./rational.js";
/** Write n = coeff²·radicand with radicand square-free (n ≥ 1). */
export declare function squareFreeFactor(n: bigint): {
    coeff: bigint;
    radicand: bigint;
};
export declare class Surd {
    readonly terms: ReadonlyMap<bigint, Rational>;
    /** square-free radicand (1n = rational part) → nonzero coefficient */
    private constructor();
    /** Drop zero coefficients so equality can stay structural. */
    private static build;
    static rational(r: Rational): Surd;
    static readonly zero: Surd;
    static readonly one: Surd;
    /** √v for a non-negative rational v, as an exact surd; undefined if v < 0. */
    static sqrt(v: Rational): Surd | undefined;
    get rationalPart(): Rational;
    isRational(): boolean;
    asRational(): Rational | undefined;
    isZero(): boolean;
    neg(): Surd;
    add(o: Surd): Surd;
    sub(o: Surd): Surd;
    mul(o: Surd): Surd;
    /** Integer power. Negative exponents need `inverse()`, so they return
     *  undefined for zero (0⁻ⁿ) or ≥2-radical bases. `x⁰ = 1` (incl. `0⁰`,
     *  matching the exact evaluator). */
    powInt(e: bigint): Surd | undefined;
    /** Multiplicative inverse, or undefined when zero or when ≥2 radicals appear. */
    inverse(): Surd | undefined;
    div(o: Surd): Surd | undefined;
    equals(o: Surd): boolean;
    toString(): string;
}
//# sourceMappingURL=surd.d.ts.map