On sharp local turns of planar polynomials

Abstract

We show that for a real polynomial of degree $n$ in two variables $x$ and $y$, any local "sharp turn" must have its "size" $\gtrsim e^{-Cn^{2}}$. We also show that there is indeed an example that has a sharp turn of size $\lesssim e^{-Cn}$. This gives a quite satisfactory answer to a problem raised by Guth. The problem was inspired by applications of the polynomial method in the study of Kakeya conjecture.