Finding roots for polynomials is a basic problem in algebra. A central topic in modern harmonic analysis is finding (sharp) estimates for various oscillatory integrals. In the 17th century, Newton developed a method, known as the Newton-Puiseux algorithm, for solving a bivariable polynomial $f(x,y) = 0$ by a fractional power series, $y = y(x^{\frac 1 m})$. In this talk, we will illustrate how one can upgrade this algorithm to prove sharp estimates for various oscillatory integrals.