From H\"older Triangles to the Whole Plane

We show how to determine whether two given real polynomial functions of a single variable are Lipschitz equivalent by comparing the values and also the multiplicities of the given polynomial functions at their critical points. Then we show how to reduce the problem of $\mathcal{R}$-semialgebraic Lipschitz classification of $\beta$-quasihomogeneous polynomials of two real variables to the problem of Lipschitz classification of real polynomial functions of a single variable, under some fairly general conditions.