On sharp local turns of planar polynomials

TL;DR

This paper investigates the limitations of sharp local turns in planar polynomials, establishing exponential bounds on their size and providing examples that nearly attain these bounds, thus addressing a problem related to the polynomial method in Kakeya conjecture research.

Contribution

It proves exponential lower bounds on the size of sharp turns in planar polynomials and constructs near-optimal examples, advancing understanding in polynomial geometry and Kakeya problem applications.

Findings

Any sharp turn in a degree n polynomial has size at least e^{-Cn^{2}}.

Existence of polynomials with sharp turns of size up to e^{-Cn}.

Addresses a problem inspired by the polynomial method in Kakeya conjecture.

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.