/**
 * Returns the number of sign changes in the polynomial coefficents after
 * applying a Mobius transformation to the given polynomial.
 *
 * * this is a specialized function used specifically by `isolateRoots`
 *
 * Applies a Mobius transformation to the given polynomial:
 * * p(x) -> (x + 1)^n * p((ax + b) / (x + 1))
 * * see e.g. https://arxiv.org/pdf/1605.00410.pdf equation (2)
 *
 * This runs in `O(n^2)` arithmetic operations (where `n` is the degree) by
 * decomposing the Mobius map into elementary steps, rather than the `O(n^3)`
 * of expanding and summing `Σ cᵢ (ax + b)^i (x + 1)^(n-i)` directly.
 *
 * The decomposition (see https://math.stackexchange.com/questions/694565)
 * uses the identity `(ax + b)/(x + 1) = a + (b - a)/(x + 1)`, which yields
 *
 *   (x + 1)^n * p((ax + b)/(x + 1)) = S₁( R( Scₐ₋ᵦ( Sₐ(p) ) ) )
 *
 * where
 *   * `Sₕ(f) = f(x + h)`           is a Taylor shift (`O(n^2)`),
 *   * `Sc_s(f)` scales the coefficient of `xⁱ` by `sⁱ` (`O(n)`),
 *   * `R(f)` reverses the coefficient array, i.e. `xⁿ f(1/x)` (`O(n)`).
 *
 * @param p a polynomial with coefficients given densely as an array of double
 * floating point numbers from highest to lowest power, e.g. `[5,-3,0]`
 * represents the polynomial `5x^2 - 3x`
 * @param p_ an error polynomial that provides a coefficient-wise error bound
 * **NOT** scaled by `γ1`
 * @param pDd a polynomial with coefficients given densely as an array of
 * double-double precision floating point numbers from highest to lowest power,
 * e.g. `[[0,5],[0,-3],[0,0]]` represents the polynomial `5x^2 - 3x`
 * @param pDd_ an array of numbers representing the absolute error bounds on the
 * coefficients of `pDd`; the actual error bound on the coefficient of `xⁱ` is `pDd_[i]*γγ(3)`
 * @param getPExact defaults to `undefined`; a function returning the exact
 * polynomial (with coefficients given as Shewchuk expansions)
 * @param a lower bound of the interval
 * @param b upper bound of the interval
 * @param A sign-certified evaluation of polynomial at the lower bound of the interval
 * @param B sign-certified evaluation of polynomial at the upper bound of the interval
 * @param failCount the number of times the Mobius transformation has failed to certify the sign
 *
 * @internal
 */
declare function mobiusAndNumSignChanges(p: number[], p_: number[], pDd: number[][], pDd_: number[], getPExact: () => number[][]): (a: number, b: number, A: number, B: number, failCount: number) => number;
declare function eMobiusAndNumSignChanges(getPExact: () => number[][], a: number, b: number, A: number, B: number): number;
export { mobiusAndNumSignChanges };
export { eMobiusAndNumSignChanges };