import { bSturmChain } from "../../../euclidean-division-related/bigint/b-sturm-chain.js";
import { bDegree } from "../../../basic/bigint/b-degree.js";
import { bSignChanges } from "./b-sign-changes.js";


/**
 * Returns the *exact* number of *distinct* real roots in the interval `(-∞,+∞)` 
 * of the given polynomial.
 * 
 * @param p a polynomial with coefficients given densely as an array of
 * bigints from highest to lowest power, e.g. `[5n,-3n,0n]`
 * represents the polynomial `5x^2 - 3x`
 * 
 * @example
 * ```typescript 
 * const p = [n1, 1n, -64n, 236n, -240n];
 * bNumRoots(p); //=> 4
 * ```
 * 
 * @doc
 */
function bNumRoots(
        p: bigint[]): number {

    const ps = bSturmChain(p);
    const as = ps.map(p => bDegree(p) % 2 === 0 ? p[0] : -p[0]);
    const bs = ps.map(p => p[0]);

    return bSignChanges(as) - bSignChanges(bs);
}


export { bNumRoots }