Rational degree is polynomially related to degree

TL;DR

This paper proves a polynomial relationship between the degree and rational degree of Boolean functions, resolving a long-standing open problem from 1994.

Contribution

It establishes that the degree of a Boolean function is at most polynomially related to its rational degree, solving a key open problem in computational complexity.

Findings

Degree is at most polynomially related to rational degree for Boolean functions

Resolved a 30-year-old open problem in complexity theory

Provides a new understanding of the relationship between different complexity measures

Abstract

We prove that $\mathrm{deg}(f) \leq \widetilde{O}(\mathrm{rdeg}(f)^3)$ for every Boolean function $f$, where $\mathrm{deg}(f)$ is the degree of $f$ and $\mathrm{rdeg}(f)$ is the rational degree of $f$. This resolves the second of the three open problems stated by Nisan and Szegedy, and attributed to Fortnow, in 1994.